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Full text of "The nature of the physical world"

Marine Biological Laboratory Library 

Woods Hole, Massachusetts 




From the estate of Eric G. Ball - 1979 



c 



J *S 




THE NATURE 
OF THE 

PHYSICAL WORLD 



THE NATURE 

OF THE 

PHYSICAL WORLD 



by 
A. S. EDDINGTON 

M.A., LL.D., D.SC, F.R.S. 

Plumian Professor of Astronomy 

in the 

University of Cambridge 



THE 

GIFFORD LECTURES 

1927 



NEW YORK : 
THE MACMILLAN COMPANY 

CAMBRIDGE, ENGLAND: 
AT THE UNIVERSITY PRESS 

I929 

All rights reserved 



Copyright, 1928, 
By THE MACMILLAN COMPANY. 



Set up and electrotyped. 
Published November, 1928. 
Reprinted February, 1929. 
Twice. March, 1929. 

Reprinted April, 1929 



SET UP BY BROWN BROTHERS LINOTYPERS 

PRINTED IN THE UNITED STATES OF AMERICA 

BY THE FERRIS PRINTING COMPANY 




PREFACE 

This book is substantially the course of Gifford Lectures 
which I delivered in the University of Edinburgh in 
January to March 1927. It treats of the philosophical 
outcome of the great changes of scientific thought which 
have recently come about. The theory of relativity and 
the quantum theory have led to strange new conceptions 
of the physical world; the progress of the principles of 
thermodynamics has wrought more gradual but no less 
profound change. The first eleven chapters are for the 
most part occupied with the new physical theories, with 
the reasons which have led to their adoption, and es- 
pecially with the conceptions which seem to underlie 
them. The aim is to make clear the scientific view of 
the world as it stands at the present day, and, where it 
is incomplete, to judge the direction in which modern 
ideas appear to be tending. In the last four chapters I 
consider the position which this scientific view should 
occupy in relation to the wider aspects of human ex- 
perience, including religion. The general spirit of the 
inquiry followed in the lectures is stated in the concluding 
paragraph of the Introduction (p. xviii). 

I hope that the scientific chapters may be read with 
interest apart from the later applications in the book; 
'but they are not written quite on the lines that would 
have been adopted had they been wholly independent. 
It would not serve my purpose to give an easy intro- 
duction to the rudiments of the relativity and quantum 
theories; it was essential to reach the later and more 
recondite developments in which the conceptions of great- 
est philosophical significance are to be found. Whilst 
much of the book should prove fairly easy reading, argu- 



vi PREFACE 

ments of considerable difficulty have to be taken in their 
turn. 

My principal aim has been to show that these scien- 
tific developments provide new material for the philoso- 
pher. I have, however, gone beyond this and indicated 
how I myself think the material might be used. I realise 
that the philosophical views here put forward can only 
claim attention in so far as they are the direct outcome 
of a study and apprehension of modern scientific work. 
General ideas of the nature of things which I may have 
formed apart from this particular stimulus from science 
are of little moment to anyone but myself. But although 
the two sources of ideas were fairly distinct in my mind 
when I began to prepare these lectures they have become 
inextricably combined in the effort to reach a coherent 
outlook and to defend it from probable criticism. For 
that reason I would like to recall that the idealistic tinge 
in my conception of the physical world arose out of math- 
ematical researches on the relativity theory. In so far as 
I had any earlier philosophical views, they were of an 
entirely different complexion. 

From the beginning I have been doubtful whether it 
was desirable for a scientist to venture so far into extra- 
scientific territory. The primary justification for such 
an expedition is that it may afford a better view of his 
own scientific domain. In the oral lectures it did not 
seem a grave indiscretion to speak freely of the various 
suggestions I had to offer. But whether they should be 
recorded permanently and given a more finished appear- 
ance has been difficult to decide. I have much to fear 
from the expert philosophical critic, but I am filled with 
even more apprehension at the thought of readers who 
may look to see whether the book is u on the side of the 
angels" and judge its trustworthiness accordingly. Dur- 



PREFACE vii 

ing the year which has elapsed since the delivery of the 
lectures I have made many efforts to shape this and other 
parts of the book into something with which I might feel 
better content. I release it now with more diffidence than 
I have felt with regard to former books. 

The conversational style of the lecture-room is gen- 
erally considered rather unsuitable for a long book, but 
I decided not to modify it. A scientific writer, in for- 
going the mathematical formulae which are his natural 
and clearest medium of expression, may perhaps claim 
some concession from the reader in return. Many parts 
of the subject are intrinsically so difficult that my only 
hope of being understood is to explain the points as I 
would were I face to face with an inquirer. 

It may be necessary to remind the American reader 
that our nomenclature for large numbers differs from 
his, so that a billion here means a million million. 

A. S. E. 
August 192S 




INTRODUCTION 

I have settled down to the task of writing these lectures 
and have drawn up my chairs to my two tables. Two 
tables! Yes; there are duplicates of every object about 
me — two tables, two chairs, two pens. 

This is not a very profound beginning to a course 
which ought to reach transcendent levels of scientific 
philosophy. But we cannot touch bedrock immediately; 
we must scratch a bit at the surface of things first. And 
whenever I begin to scratch the first thing I strike is — 
my two tables. 

One of them has been familiar to me from earliest 
years. It is a commonplace object of that environment 
which I call the world. How shall I describe it? It has 
extension; it is comparatively permanent; it is coloured; 
above all it is substantial. By substantial I do not merely 
mean that it does not collapse when I lean upon it ; I mean 
that it is constituted of "substance" and by that word 
I am trying to convey to you some conception of its 
intrinsic nature. It is a thing; not like space, which is 
a mere negation; nor like time, which is — Heaven 
knows what ! But that will not help you to my meaning 
because it is the distinctive characteristic of a "thing" 
to have this substantiality, and I do not think substan- 
tiality can be described better than by saying that it is 
the kind of nature exemplified by an ordinary table. And 
so we go round in circles.^ After all if you are a plain 
commonsense man, not too much worried with scien- 
tific scruples, you will be confident that you understand 
the nature of an ordinary table. I have even heard 
of plain men who had the idea that they could better 
understand the mystery of their own nature if scientists 

ix 



x INTRODUCTION 

would discover a way of explaining it in terms of the 
easily comprehensible nature of a table. 

Table No. 2 is my scientific table. It is a more recent 
acquaintance and I do not feel so familiar with it. It 
does not belong to the world previously mentioned — 
that world which spontaneously appears around me when 
I open my eyes, though how much of it is objective and 
how much subjective I do not here consider. It is part 
of a world which in more devious ways has forced 
itself on my attention. My scientific table is mostly 
emptiness. Sparsely scattered in that emptiness are ! 
numerous electric charges rushing about with great 
speed; but their combined bulk amounts to less than a 
billionth of the bulk of the table itself. Notwithstanding 
its strange construction it turns out to be an entirely 
efficient table. It supports my writing paper as satisfac- 
torily as table No. 1 ; for when I lay the paper on it the 
little electric particles with their headlong speed keep 
on hitting the underside, so that the paper is maintained 
in shuttlecock fashion at a nearly steady level. If I lean 
upon this table I shall not go through; or, to be strictly 
accurate, the chance of my scientific elbow going through 
my scientific table is so excessively small that it can be 
neglected in practical life. Reviewing their properties 
one by one, there seems to be nothing to choose between 
the two tables for ordinary purposes; but when ab- 
normal circumstances befall, then my scientific table 
shows to advantage. If the house catches fire my sci- 
entific table will dissolve quite naturally into scientific 
smoke, whereas my familiar table undergoes a metamor- 
phosis of its substantial nature which I can only regard 
as miraculous. 

There is nothing substantial about my second table. 
It is nearly all empty space — space pervaded, it is true, 



INTRODUCTION xi 

by fields of force, but these are assigned to the category 
of "influences", not of "things". Even in the minute 
part which is not empty we must not transfer the old 
notion of substance. In dissecting matter into electric 
charges we have travelled far from that picture of it 
which first gave rise to the conception of substance, and 
the meaning of that conception — if it ever had any — 
has been lost by the way. The whole trend of modern 
scientific views is to break down the separate categories 
of "things", "influences", "forms", etc., and to substi- 
tute a common background of all experience. Whether 
we are studying a material object, a magnetic field, a 
geometrical figure, or a duration of time, our scientific 
information is summed up in measures ; neither the appa- 
ratus of measurement nor the mode of using it suggests 
that there is anything essentially different in these prob- 
lems. The measures themselves afford no ground for 
a classification by categories. We feel it necessary to 
concede some background to the measures — an external 
world; but the attributes of this world, except in so far 
as they 'are reflected in the measures, are outside scien- 
tific scrutiny. Science has at last revolted against 
attaching the exact knowledge contained in these meas- 
urements to a traditional picture-gallery of conceptions 
which convey no authentic information of the back- 
ground and obtrude irrelevancies into the scheme of 
knowledge. 

I will not here stress .further the non-substantiality 
of electrons, since it is scarcely necessary to the present 
line of thought. Conceive them as substantially as you 
will, there is a vast difference between my scientific table 
with its substance (if any) thinly scattered in specks 
in a region mosdy empty and the table of everyday 
conception which we regard as the type of solid reality 



xii INTRODUCTION 

— an incarnate protest against Berkleian subjectivism. 
It makes all the difference in the world whether the 
paper before me is poised as it were on a swarm of flies 
and sustained in shuttlecock fashion by a series of tiny 
blows from the swarm underneath, or whether it is sup- 
ported because there is substance below it, it being the 
intrinsic nature of substance to occupy space to the exclu- 
sion of other substance; all the difference in conception 
at least, but no difference to my practical task of writing 
on the paper. 

I need not tell you that modern physics has by deli- 
cate test and remorseless logic assured me that my sec- 
ond scientific table is the only one which is really there — 
wherever "there" may be. On the other hand I need 
not tell you that modern physics will never succeed in 
exorcising that first table — strange compound of external 
nature, mental imagery and inherited prejudice — which 
lies visible to my eyes and tangible to my grasp. We 
must bid good-bye to it for the present for we are about 
to turn from the familiar world to the scientific world 
revealed by physics. This is, or is intended to be, a 
wholly external world. 

"You speak paradoxically of two worlds. Are they 
not really two aspects or two interpretations of one and 
the same world?" 

Yes, no doubt they are ultimately to be identified 
after some fashion. But the process by which the ex- 
ternal world of physics is transformed into a world of 
familiar acquaintance in human consciousness is outside 
the scope of physics. And so the world studied accord- 
ing to the methods of physics remains detached from 
the world familiar to consciousness, until after the 
physicist has finished his labours upon it. Provisionally, 
therefore, we regard the table which is the subject of 



INTRODUCTION xiii 

physical research as altogether separate from the familiar 
table, without prejudging the question of their ultimate 
identification. It is true that the whole scientific 
inquiry starts from the familiar world and in the end it 
must return to the familiar world; but the part of the 
journey over which the physicist has charge is in foreign 
territory. 

Until recently there was a much closer linkage; the 
physicist used to borrow the raw material of his world 
from the familiar world, but he does so no longer. His 
raw materials are aether, electrons, quanta, potentials, 
Hamiltonian functions, etc., and he is nowadays scrupu- 
lously careful to guard these from contamination by con- 
ceptions borrowed from the other world. There is a 
familiar table parallel to the scientific table, but there is 
no familiar electron, quantum or potential parallel to the 
scientific electron, quantum or potential. We do not even 
desire to manufacture a familiar counterpart to these 
things or, as we should commonly say, to "explain" the 
electron. After the physicist has quite finished his world- 
building a linkage or identification is allowed; but prema- 
ture attempts at linkage have been found to be entirely 
mischievous. 

Science aims at constructing a world which shall be 
symbolic of the world of commonplace experience. It 
is not at all necessary that every individual symbol that 
is used should represent something in common experi- 
ence or even something explicable in terms of com- 
mon experience. The man in the street is always mak- 
ing this demand for concrete explanation of the things 
referred to in science; but of necessity he must be 
disappointed. It is like our experience in learning to 
read. That which is written in a book is symbolic of a 
story in real life. The whole intention of the book is 



xiv INTRODUCTION 

that ultimately a reader will identify some symbol, say 
BREAD, with one of the conceptions of familiar life. But 
it is mischievous to attempt such identifications prema- 
turely, before the letters are strung into words and 
the words into sentences. The symbol A is not the 
counterpart of anything in familiar life. To the child 
the letter A would seem horribly abstract; so we give 
him a familiar conception along with it. "A was an 
Archer who shot at a frog." This tides over his imme- 
diate difficulty; but he cannot make serious progress with 
word-building so long as Archers, Butchers, Captains, 
dance round the letters. The letters are abstract, and 
sooner or later he has to realise it. In physics we have 
outgrown archer and apple-pie definitions of the funda- 
mental symbols. To a request to explain what an electron 
really is supposed to be we can only answer, "It is part 
of the A B c of physics". 

The external world of physics has thus become a world 
of shadows. In removing our illusions we have removed 
the substance, for indeed we have seen that substance is 
one of the greatest of our illusions. Later perhaps 
we may inquire whether in our zeal to cut out all that is 
unreal we may not have used the knife too ruthlessly. 
Perhaps, indeed, reality is a child which cannot survive 
without its nurse illusion. But if so, that is of little con- 
cern to the scientist, who has good and sufficient reasons 
for pursuing his investigations in the world of shadows 
and is content to leave to the philosopher the determina- 
tion of its exact status in regard to reality. In the world 
of physics we watch a shadowgraph performance of 
the drama of familiar life. The shadow of my 
elbow rests on the shadow table as the shadow ink 
flows over the shadow paper. It is all symbolic, and 
as a symbol the physicist leaves it. Then comes the 



INTRODUCTION xv 

alchemist Mind who transmutes the symbols. The 
sparsely spread nuclei of electric force become a tangible 
solid; their restless agitation becomes the warmth of 
summer; the octave of aethereal vibrations becomes a 
gorgeous rainbow. Nor does the alchemy stop here. In 
the transmuted world new significances arise which are 
scarcely to be traced in the world of symbols; so that 
it becomes a world of beauty and purpose — and, alas, suf- 
fering and evil. 

The frank realisation that physical science is con- 
cerned with a world of shadows is one of the most sig- 
nificant of recent advances. I do not mean that physicists 
are to any extent preoccupied with the philosophical impli- 
cations of this. From their point of view it is not so much 
a withdrawal of untenable claims as an assertion of free- 
dom for autonomous development. At the moment I am 
not insisting on the shadowy and symbolic character of 
the world of physics because of its bearing on philosophy, 
but because the aloofness from familiar conceptions will 
be apparent in the scientific theories I have to describe. 
If you are not prepared for this aloofness you are 
likely to be out of sympathy with modern scientific 
theories, and may even think them ridiculous — as, I 
daresay, many people do. 

It is difficult to school ourselves to treat the physical 
world as purely symbolic. We are always relapsing and 
mixing with the symbols incongruous conceptions taken 
from the world of consciousness. Untaught by long 
experience we stretch a hand to grasp the shadow, 
instead of accepting its shadowy nature. Indeed, unless 
we confine ourselves altogether to mathematical sym- 
bolism it is hard to avoid dressing our symbols in deceit- 
ful clothing. When I think of an electron there 
rises to my mind a hard x red, tiny ball; the proton simi- 



xvi INTRODUCTION 

larly is neutral grey. Of course the colour is absurd — 
perhaps not more absurd than the rest of the conception — 
but I am incorrigible. I can well understand that the 
younger minds are finding these pictures too concrete 
and are striving to construct the world out of Hamil- 
tonian functions and symbols so far removed from 
human preconception that they do not even obey 
the laws of orthodox arithmetic. For myself I find some 
difficulty in rising to that plane of thought; but I am 
convinced that it has got to come. 

In these lectures I propose to discuss some of the 
results of modern study of the physical world which 
give most food for philosophic thought. This will include 
new conceptions in science and also new knowledge. In 
both respects we are led to think of the material uni- 
verse in a way very different from that prevailing at the 
end of the last century. I shall not leave out of 
sight the ulterior object which must be in the mind of 
a Gifford Lecturer, the problem of relating these 
purely physical discoveries to the wider aspects and 
interests of our human nature. These relations can- 
not but have undergone change, since our whole concep- 
tion of the physical world has radically changed. I am 
convinced that a just appreciation of the physical 
world as it is understood to-day carries with it a feeling 
of open-mindedness towards a wider significance tran- 
scending scientific measurement, which might have 
seemed illogical a generation ago; and in the later 
lectures I shall try to focus that feeling and make 
inexpert efforts to find where it leads. But I should 
be untrue to science if I did not insist that its study is 
an end in itself. The path of science must be pursued 
for its own sake, irrespective of the views it may afford 
of a wider landscape; in this spirit we must follow the 



INTRODUCTION xvii 

path whether it leads to the hill of vision or the tunnel 
of obscurity. Therefore till the last stage of the course 
is reached you must be content to follow with me the 
beaten track of science, nor scold me too severely for 
loitering among its wayside flowers. That is to be the 
understanding between us. Shall we set forth? 



CONTENTS 

Preface vii 

Introduction xi 

Chapter I. The Downfall of Classical Physics I 

II. Relativity 20 

III. Time 36 

IV. The Running-Down of the Universe 63 
V. "Becoming" 87 

VI. Gravitation — the Law 111 

VII. Gravitation — the Explanation 138 

VIII. Man's Place in the Universe 163 

IX. The Quantum Theory 179 

X. The New Quantum Theory 200 

XI. World Building 230 

XII. Pointer Readings 247 

XIII. Reality . 273 

XIV. Causation 293 
XV. Science and Mysticism 316 

Conclusion 343 

Index 355 



THE NATURE 
OF THE 

PHYSICAL WORLD 



Chapter I 

THE DOWNFALL OF CLASSICAL PHYSICS 

The Structure of the Atom. Between 1905 and 1908 Ein- 
stein and Minkowski introduced fundamental changes in 
our ideas of time and space. In 191 1 Rutherford intro- 
duced the greatest change in our idea of matter since the 
time of Democritus. The reception of these two changes 
was curiously different. The new ideas of space and time 
were regarded on all sides as revolutionary; they were 
received with the greatest enthusiasm by some and 
the keenest opposition by others. The new idea of mat- 
ter underwent the ordinary experience of scientific dis- 
covery; it gradually proved its worth, and when the 
evidence became overwhelmingly convincing it quietly 
supplanted previous theories. No great shock was felt. 
And yet when I hear to-day protests against the Bolshev- 
ism of modern science and regrets for the old-established 
order, I am inclined to think that Rutherford, not Ein- 
stein, is the real villain of the piece. When we compare 
the universe as it is now supposed to be with the universe 
as we had ordinarily preconceived it, the most arresting 
change is not the rearrangement of space and time by 
Einstein but the dissolution of all that we regard as most 
solid into tiny specks floating in void. That gives an 
abrupt jar to those who think that things are more or 
less what they seem. The revelation by modern physics 
of the void within the atom is more disturbing than 
the revelation by astronomy of the immense void of 
interstellar space. 

The atom is as porous as the solar system. If we 
eliminated all the unfilled space in a man's body and 



2 DOWNFALL OF CLASSICAL PHYSICS 

collected his protons and electrons into one mass, the 
man would be reduced to a speck just visible with a 
magnifying glass. 

This porosity of matter was not foreshadowed in the 
atomic theory. Certainly it was known that in a gas 
like air the atoms are far separated, leaving a great deal 
of empty space; but it was only to be expected that mate- 
rial with the characteristics of air should have rela- 
tively little substance in it, and "airy nothing" is a com- 
mon phrase for the insubstantial. In solids the atoms 
are packed tightly in contact, so that the old atomic 
theory agreed with our preconceptions in regard- 
ing solid bodies as mainly substantial without much 
interstice. 

The electrical theory of matter which arose towards 
the end of the nineteenth century did not at first alter 
this view. It was known that the negative electricity 
was concentrated into unit charges of very small bulk; 
but the other constituent of matter, the positive elec- 
tricity, was pictured as a sphere of jelly of the same 
dimensions as the atom and having the tiny negative 
charges embedded in it. Thus the space inside a solid 
was still for the most part well filled. 

But in 191 1 Rutherford showed that the positive 
electricity was also concentrated into tiny specks. His 
scattering experiments proved that the atom was able to 
exert large electrical forces which would be impossible 
unless the positive charge acted as a highly concentrated 
source of attraction; it must be contained in a nucleus 
minute in comparison with the dimensions of the atom. 
Thus for the first time the main volume of the atom was 
entirely evacuated, and a "solar system" type of atom 
was substituted for a substantial "billiard-ball". Two 
years later Niels Bohr developed his famous theory on 



THE STRUCTURE OF THE ATOM 3 

the basis of the Rutherford atom, and since then rapid 
progress has been made. Whatever further changes of 
view are in prospect, a reversion to the old substantial 
atoms is unthinkable. 

The accepted conclusion at the present day is that all 
varieties of matter are ultimately composed of two ele- 
mentary constituents — protons and electrons. Electrically 
these are the exact opposites of one another, the proton 
being a charge of positive electricity and the electron 
a charge of negative electricity. But in other respects 
their properties are very different. The proton has 1840 
times the mass of the electron, so that nearly all the 
mass of matter is due to its constituent protons. 
The proton is not found unadulterated except in hydro- 
gen, which seems to be the most primitive form of mat- 
ter, its atom consisting of one proton and one electron. 
In other atoms a number of protons and a lesser 
number of electrons are cemented together to form 
a nucleus; the electrons required to make up the bal- 
ance are scattered like remote satellites of the nucleus, 
and can even escape from the atom and wander freely 
through the material. The diameter of an electron is 
about 1/50,000 of the diameter of an atom; that of the 
nucleus is not very much larger; an isolated proton is 
supposed to be much smaller still. 

Thirty years ago there was much debate over the ques- 
tion of aether-drag — whether the earth moving round 
the sun drags the aether with it. At that time the solidity 
of the atom was unquestioned, and it was difficult 
to believe that matter could push its way through the 
aether without disturbing it. It was surprising and per- 
plexing to find as the result of experiments that no 
convection of the aether occurred. But we now realise 
that the aether can slip through the atoms as easily as 



4 DOWNFALL OF CLASSICAL PHYSICS 

through the solar system, and our expectation is all the 
other way. 

We shall return to the "solar system" atom in later 
chapters. For the present the two things which concern 
us are (i) its extreme emptiness, and (2) the fact that it 
is made up of electrical charges. 

Rutherford's nuclear theory of the atom is not usually 
counted as one of the scientific revolutions of the present 
century. It was a far-reaching discovery, but a discovery 
falling within the classical scheme of physics. The nature 
and significance of the discovery could be stated in plain 
terms, i.e. in terms of conceptions already current in 
science. The epithet "revolutionary" is usually reserved 
for two great modern developments — the Relativity 
Theory and the Quantum Theory. These are not 
merely new discoveries as to the content of the world; 
they involve changes in our mode of thought about the 
world. They cannot be stated immediately in plain 
terms because we have first to grasp new conceptions 
undreamt of in the classical scheme of physics. 

I am not sure that the phrase "classical physics" has 
ever been closely defined. But the general idea is that 
the scheme of natural law developed by Newton in the 
Principia provided a pattern which all subsequent devel- 
opments might be expected to follow. Within the four 
corners of the scheme great changes of outlook were 
possible; the wave-theory of light supplanted the cor- 
puscular theory; heat was changed from substance (calo- 
ric) to energy of motion; electricity from continuous 
fluid to nuclei of strain in the aether. But this was all 
allowed for in the elasticity of the original scheme. 
Waves, kinetic energy, and strain already had their 
place in the scheme; and the application of the same 
conceptions to account for a wider range of phenomena 



THE FITZGERALD CONTRACTION 5 

was a tribute to the comprehensiveness of Newton's 
original outlook. 

We have now to see how the classical scheme broke 
down. 

The FitzGerald Contraction. We can best start from 
the following fact. Suppose that you have a rod moving 
at very high speed. Let it first be pointing transverse 
to its line of motion. Now turn it through a right angle 
so that it is along the line of motion. The rod contracts. 
It is shorter when it is along the line of motion than 
when it is across the line of motion. 

This contraction, known as the FitzGerald contrac- 
tion, is exceedingly small in all ordinary circumstances. 
It does not depend at all on the material of the rod but 
only on the speed. For example, if the speed is 19 miles 
a second — the speed of the earth round the sun — the 
contraction of length is 1 part in 200,000,000, or 2^4 
inches in the diameter of the earth. 

This is demonstrated by a number of experiments of 
different kinds of which the earliest and best known is 
the Michelson-Morley experiment first performed in 
1887, repeated more accurately by Morley and Miller 
in 1905, and again by several observers within the last 
year or two. I am not going to describe these experi- 
ments except to mention that the convenient way of 
giving your rod a large velocity is to carry it on the 
earth which moves at high^ speed round the sun. Nor 
shall I discuss here how complete is the proof afforded 
by these experiments. It is much more important that 
you should realise that the contraction is just what would 
be expected from our current knowledge of a material 
rod. 

You are surprised that the dimensions of a moving, 



6 DOWNFALL OF CLASSICAL PHYSICS 

rod can be altered merely by pointing it different ways. 
You expect them to remain unchanged. But which rod 
are you thinking of? (You remember my two tables.) 
If you are thinking of continuous substance, extending in 
space because it is the nature of substance to occupy 
space, then there seems to be no valid cause for a change 
of dimensions. But the scientific rod is a swarm of 
electrical particles rushing about and widely separated 
from one another. The marvel is that such a swarm 
should tend to preserve any definite extension. The 
particles, however, keep a certain average spacing so 
that the whole volume remains practically steady; they 
exert electrical forces on one another, and the volume 
which they fill corresponds to a balance between the 
forces drawing them together and the diverse motions 
tending to spread them apart. When the rod is set in 
motion these electrical forces change. Electricity in 
motion constitutes an electric current. But electric 
currents give rise to forces of a different type from those 
due to electricity at rest, viz. magnetic forces. More- 
over these forces arising from the motion of electric 
charges will naturally be of different intensity in the 
directions along and across the line of motion. 

By setting in motion the rod with all the little electric 
charges contained in it we introduce new magnetic forces 
between the particles. Clearly the original balance is 
upset, and the average spacing between the particles 
must alter until a new balance is found. And so the 
extension of the swarm of particles — the length of the 
rod — alters. 

There is really nothing mysterious about the Fitz- 
Gerald contraction. It would be an unnatural property 
of a rod pictured in the old way as continuous substance 
occupying space in virtue of its substantiality; but it is 



THE FITZGERALD CONTRACTION 7 

an entirely natural property of a swarm of particles held 
in delicate balance by electromagnetic forces, and occu- 
pying space by buffeting away anything that tries to 
enter. Or you may look at it this way: your expecta- 
tion that the rod will keep its original length presup- 
poses, of course, that it receives fair treatment and 
is not subjected to any new stresses. But a rod in motion 
is subjected to a new magnetic stress, arising not from 
unfair outside tampering but as a necessary consequence 
of its own electrical constitution; and under this stress 
the contraction occurs. Perhaps you will think that if 
the rod were rigid enough it might be able to resist the 
compressing force. That is not so; the FitzGerald con- 
traction is the same for a rod of steel and for a rod of 
india-rubber; the rigidity and the compressing stress are 
bound up with the constitution in such a way that if 
one is large so also is the other. It is necessary to rid 
our minds of the idea that this failure to keep a constant 
length is an imperfection of the rod; it is only imperfect 
as compared with an imaginary "something" which has 
not this electrical constitution — and therefore is not 
material at all. The FitzGerald contraction is not an 
imperfection but a fixed and characteristic property of 
matter, like inertia. 

We have here drawn a qualitative inference from the 
electrical structure of matter; we must leave it to the 
mathematician to calculate the quantitative effect. The 
problem was worked out by Lorentz and Larmor about 
1900. They calculated the change in the average spacing 
of the particles required to restore the balance after it 
had been upset by the new forces due to the change of 
motion of the charges. This calculation was found to 
give precisely the FitzGerald contraction, i.e. the amount 
already inferred from the experiments above mentioned. 



8 DOWNFALL OF CLASSICAL PHYSICS 

Thus we have two legs to stand on. Some will prefer to 
trust the results because they seem to be well established 
by experiment; others will be more easily persuaded by 
the knowledge that the FitzGerald contraction is a 
necessary consequence of the scheme of electromag- 
netic laws universally accepted since the time of Max- 
well. Both experiments and theories sometimes go 
wrong; so it is just as well to have both alternatives. 

Consequences of the Contraction. This result alone, 
although it may not quite lead you to the theory of rela- 
tivity, ought to make you uneasy about classical physics. 
The physicist when he wishes to measure a length — 
and he cannot get far in any experiment without meas- 
uring a length — takes a scale and turns it in the direc- 
tion needed. It never occurred to him that in spite 
of all precautions the scale would change length when 
he did this; but unless the earth happens to be at rest 
a change must occur. The constancy of a measur- 
ing scale is the rock on which the whole structure of 
physics has been reared; and that rock has crumbled 
away. You may think that this assumption cannot have 
betrayed the physicist very badly; the changes of length 
cannot be serious or they would have been noticed. 
Wait and see. 

Let us look at some of the consequences of the Fitz- 
Gerald contraction. First take what may seem to be a 
rather fantastic case. Imagine you are on a planet mov- 
ing very fast indeed, say 161,000 miles a second. For 
this speed the contraction is one-half. Any solid con- 
tracts to half its original length when turned from across 
to along the line of motion. A railway journey between 
two towns which was 100 miles at noon is shortened to 
50 miles at 6 p.m. when the planet has turned through 



CONSEQUENCES OF THE CONTRACTION 9 

a right angle. The inhabitants copy Alice in Wonder- 
land; they pull out and shut up like a telescope. 

I do not know of a planet moving at 161,000 miles 
a second, but I could point to a spiral nebula far away 
in space which is moving at 1000 miles a second. This 
may well contain a planet and (speaking unprofession- 
ally) perhaps I shall not be taking too much licence if 
I place intelligent beings on it. At 1000 miles a second 
the contraction is not large enough to be appreciable in 
ordinary affairs; but it is quite large enough to be appre- 
ciable in measurements of scientific or even of engi- 
neering accuracy. One of the most fundamental pro- 
cedures in physics is to measure lengths with a scale 
moved about in any way. Imagine the consternation of 
the physicists on this planet when they learn that they 
have made a mistake in supposing that their scale is a 
constant measure of length. What a business to go back 
over all the experiments ever performed, apply the 
corrections for orientation of the scale at the time, and 
then consider de novo the inferences and system of 
physical laws to be deduced from the amended data ! 
How thankful our own physicists ought to be that they 
are not in this runaway nebula but on a decently slow- 
moving planet like the earth ! 

But stay a moment. Is it so certain that we are on 
a slow-moving planet? I can imagine the astronomers 
in that nebula observing far away in space an insignifi- 
cant star attended by an insignificant planet called 
Earth. They observe too that it is moving with the 
huge velocity of 1000 miles a second; because naturally 
if we see them receding from us at 1000 miles a second 
they will see us receding from them at 1000 miles a 
second. "A thousand miles a second!" exclaim the 
nebular physicists, "How unfortunate for the poor 



io DOWNFALL OF CLASSICAL PHYSICS 

physicists on the Earth! The FitzGerald contraction 
will be quite appreciable, and all their measures with 
scales will be seriously wrong. What a weird system of 
laws of Nature they will have deduced, if they have over- 
looked this correction !" 

There is no means of deciding which is right — to 
which of us the observed relative velocity of iooo 
miles a second really belongs. Astronomically the gal- 
axy of which the earth is a member does not seem to 
be more important, more central, than the nebula. 
The presumption that it is we who are the more nearly 
at rest has no serious foundation; it is mere self- 
flattery. 

"But", you will say, "surely if these appreciable 
changes of length occurred on the earth, we should 
detect them by our measurements." That brings me to 
the interesting point. We could not detect them by any 
measurement; they may occur and yet pass quite un- 
noticed. Let me try to show how this happens. 

This room, we will say, is travelling at 161,000 miles 
a second vertically upwards. That is my statement, and 
it is up to you to prove it wrong. I turn my arm from 
horizontal to vertical and it contracts to half its original 
length. You don't believe me? Then bring a yard- 
measure and measure it. First, horizontally, the result 
is 30 inches; now vertically, the result is 30 half-inches. 
You must allow for the fact that an inch-division of the 
scale contracts to half an inch when the yard-measure 
is turned vertically. 

"But we can see that your arm does not become 
shorter; can we not trust our own eyes?" 

Certainly not, unless you remember that when you 
got up this morning your retina contracted to half its 
original width in the vertical direction; consequently it 



CONSEQUENCES OF THE CONTRACTION n 

is now exaggerating vertical distances to twice the scale 
of horizontal distances. 

"Very well", you reply, "I will not get up. I will lie 
in bed and watch you go through your performance in 
an inclined mirror. Then my retina will be all right, 
but I know I shall still see no contraction." 

But a moving mirror does not give an undistorted 
image of what is happening. The angle of reflection of 
light is altered by motion of a mirror, just as the angle 
of reflection of a billiard-ball would be altered if the 
cushion were moving. If you will work out by the 
ordinary laws of optics the effect of moving a mirror at 
161,000 miles a second, you will find that it introduces 
a distortion which just conceals the contraction of my 
arm. 

And so on for every proposed test. You cannot 
disprove my assertion, and, of course, I cannot prove 
it; I might equally well have chosen and defended any 
other velocity. At first this seems to contradict what 
I told you earlier — that the contraction. had been proved 
and measured by the Michelson-Morley and other experi- 
ments — but there is really no contradiction. They were 
all null experiments, just as your experiment of watch- 
ing my arm in an inclined mirror was a null experiment. 
Certain optical or electrical consequences of the earth's 
motion were looked for of the same type as the 
distortion of images by a moving mirror; these would 
have been observed unless a contraction occurred of 
just the right amount to compensate them. They 
were not observed; therefore the compensating contrac- 
tion had occurred. There was just one alternative; the 
earth's true velocity through space might happen to 
have been nil. This was ruled out by repeating the 
experiment six months later, since the earth's motion 



12 DOWNFALL OF CLASSICAL PHYSICS 

could not be nil on both occasions. Thus the contraction 
was demonstrated and its law of dependence on velocity 
verified. But the actual amount of contraction on either 
occasion was unknown, since the earth's true velocity 
(as distinct from its orbital velocity with respect to the 
sun) was unknown. It remains unknown because the 
optical and electrical effects by which we might hope 
to measure it are always compensated by the contraction. 
I have said that the constancy of a measuring scale is 
the rock on which the structure of physics has been 
reared. The structure has also been supported by sup- 
plementary props because optical and electrical devices 
can often be used instead of material scales to ascertain 
lengths and distances. But we find that all these are 
united in a conspiracy not to give one another away. 
The rock has crumbled and simultaneously all the other 
supports have collapsed. 

Frames of Space. We can now return to the quarrel 
between the nebular physicists and ourselves. One of us 
has a large velocity and his scientific measurements are 
seriously affected by the contraction of his scales. Each 
has hitherto taken it for granted that it is the other 
fellow who is making the mistake. We cannot settle 
the dispute by appeal to experiment because in every 
experiment the mistake introduces two errors which just 
compensate one another. 

It is a curious sort of mistake which always carries 
with it its own compensation. But remember that the 
compensation only applies to phenomena actually ob- 
served or capable of observation. The compensation 
does not apply to the intermediate part of our deduc- 
tion — that system of inference from observation which 
forms the classical physical theory of the universe. 



FRAMES OF SPACE 13 

Suppose that we and the nebular physicists survey 
the world, that is to say we allocate the surrounding 
objects to their respective positions in space. One 
party, say the nebular physicists, has a large velocity; 
their yard-measures will contract and become less than 
a yard when they measure distances in a certain direc- 
tion; consequently they will reckon distances in that 
direction too great. It does not matter whether they 
use a yard-measure, or a theodolite, or merely judge 
distances with the eye; all methods of measurement 
must agree. If motion caused a disagreement of any 
kind, we should be able to determine the motion by 
observing the amount of disagreement; but, as we have 
already seen, both theory and observation indicate that 
there is complete compensation. If the nebular physi- 
cists try to construct a square they will construct an 
oblong. No test can ever reveal to them that it is not a 
square; the greatest advance they can make is to recog- 
nise that there are people in another world who have got 
it into their heads that it is an oblong, and they may be 
broadminded enough to admit that this point of view, ab- 
surd as it seems, is really as defensible as their own. It 
is clear that their whole conception of space is distorted 
as compared with ours, and ours is distorted as com- 
pared with theirs. We are regarding the same universe, 
but we have arranged it in different spaces. The original 
quarrel as to whether they or we are moving with the 
speed of 1000 miles a second has made so deep a cleavage 
between us that we cannot even use the same space. 

Space and time are words conveying more than one 
meaning. Space is an empty void; or it is such and such 
a number of inches, acres, pints. Time is an ever-rolling 
stream; or it is something signalled to us by wireless. 
The physicist has no use for vague conceptions; he often 



i 4 DOWNFALL OF CLASSICAL PHYSICS 

has them, alas ! but he cannot make real use of them. 
So when he speaks of space it is always the inches or 
pints that he should have in mind. It is from this point 
of view that our space and the space of the nebular 
physicists are different spaces; the reckoning of inches 
and pints is different. To avoid possible misunder- 
standing it is perhaps better to say that we have different 
frames of space — different frames to which we refer the 
location of objects. Do not, however, think of a frame 
of space as something consciously artificial; the frame 
of space comes into our minds with our first perception of 
space. Consider, for example, the more extreme case 
when the FitzGerald contraction is one-half. If a man 
takes a rectangle 2"Xi" to be a square it is clear that 
space must have dawned on his intelligence in a way very 
different from that in which we have apprehended it. 

The frame of space used by an observer depends only 
on his motion. Observers on different planets with the 
same velocity (i.e. having zero relative velocity) will 
agree as to the location of the objects of the universe; 
but observers on planets with different velocities have 
different frames of location. You may ask, How can 
I be so confident as to the way in which these imaginary 
beings will interpret their observations? If that objec- 
tion is pressed I shall not defend myself; but those who 
dislike my imaginary beings must face the alternative 
of following the argument with mathematical symbols. 
Our purpose has been to express in a conveniently 
apprehensible form certain results which follow from 
terrestrial experiments and calculations as to the effect 
of motion on electrical, optical and metrical phenomena. 
So much careful work has been done on this subject 
that science is in a position to state what will be the 
consequence of making measurements with instruments 



FRAMES OF SPACE 15 

travelling at high speed — whether instruments of a 
technical kind or, for example, a human retina. In only 
one respect do I treat my nebular observer as more than 
a piece of registering apparatus; I assume that he is 
subject to a common failing of human nature, viz. he 
takes it for granted that it was his planet that God 
chiefly had in mind when the universe was created. 
Hence he is (like my reader perhaps?) disinclined to 
take seriously the views of location of those people who 
are so misguided as to move at 1000 miles a second 
relatively to his parish pump. 

An exceptionally modest observer might take some 
other planet than his own as the standard of rest. Then 
he would have to correct all his measurements for the 
FitzGerald contraction due to his own motion with 
respect to the standard, and the corrected measures 
would give the space-frame belonging to the standard 
planet as the original measures gave the space-frame of 
his own planet. For him the dilemma is even more 
pressing, for there is nothing to guide him as to the 
planet to be selected for the standard of rest. Once 
he gives up the naive assumption that his own frame is 
the one and only right frame the question arises, Which 
then of the innumerable other frames is right? There 
is no answer, and so far as we can see no possibility of 
an answer. Meanwhile all his experimental measure- 
ments are waiting unreduced, because the corrections 
to be applied to them depend on the answer. I am 
afraid our modest observer will get rather left behind 
by his less humble colleagues. 

The trouble that arises is not that we have found 
anything necessarily wrong with the frame of location 
that has been employed in our system of physics; it has 
not led to experimental contradictions. The only thing 



16 DOWNFALL OF CLASSICAL PHYSICS 

known to be "wrong" with it is that it is not unique. 
If we had found that our frame was unsatisfactory and 
another frame was preferable, that would not have 
caused a great revolution of thought; but to discover 
that ours is one of many frames, all of which are equally 
satisfactory, leads to a change of interpretation of the 
significance of a frame of location. 

"Commonsense" Objections. Before going further I must 
answer the critic who objects in the name of common- 
sense. Space — his space — is so vivid to him. "This 
object is obviously here; that object is just there. I know 
it; and I am not going to be shaken by any amount of sci- 
entific obscurantism about contraction of measuring rods." 
We have certain preconceived ideas about location 
in space which have come down to us from ape-like 
ancestors. They are deeply rooted in our mode of 
thought, so that it is very difficult to criticise them 
impartially and to realise the very insecure foundation 
on which they rest. We commonly suppose that each 
of the objects surrounding us has a definite location in 
space and that we are aware of the right location. The 
objects in my study are actually in the positions where 
I am "aware" that they are; and if an observer (on 
another star) surveying the room with measuring rods, 
etc., makes out a different arrangement of location, he 
is merely spinning a scientific paradox which does not 
shake the real facts of location obvious to any man 
of commonsense. This attitude rejects with contempt 
the question, How am I aware of the location? If the 
location is determined by scientific measurements with 
elaborate precautions, we are ready enough to sug- 
gest all sorts of ways in which the apparatus might 
have misbehaved; but if the knowledge of location is 



a 



COMMONSENSE" OBJECTIONS 17 



obtained with no precautions, if it just comes into our 
heads unsought, then it is obviously true and to doubt 
it would be flying in the face of commonsense ! We 
have a sort of impression (although we do not like 
to acknowledge it) that the mind puts out a feeler into 
space to ascertain directly where each familiar object is. 
That is nonsense; our commonsense knowledge of location 
is not obtained that way. Strictly it is sense knowledge, 
not commonsense knowledge. It is partly obtained 
by touch and locomotion; such and such an object 
is at arm's length or a few steps away. Is there 
any essential difference (other than its crudity) between 
this method and scientific measurements with a scale? 
It is partly obtained by vision — a crude version of 
scientific measurement with a theodolite. Our common 
knowledge of where things are is not a miraculous 
revelation of unquestionable authority; it is inference 
from observations of the same kind as, but cruder than, 
those made in a scientific survey. Within its own limits 
of accuracy the scheme of location of objects that I am 
instinctively "aware" of is the same as my scientific 
scheme of location, or frame of space. 

When we use a carefully made telescope lens and a 
sensitised plate instead of the crystalline lens and retina 
of the eye we increase the accuracy but do not alter the 
character of our survey of space. It is by this increase 
of refinement that we have become "aware" of certain 
characteristics of space which were not known to our 
ape-like ancestor when he instituted the common ideas 
that have come down to us. His scheme of location 
works consistently so long as there is no important 
change in his motion (a few miles a second makes no 
appreciable difference) ; but a large change involves a 
transition to a different system of location which is like- 



18 DOWNFALL OF CLASSICAL PHYSICS 

wise self-consistent, although it is inconsistent with the 
original one. Having any number of these systems of 
location, or frames of space, we can no longer pretend 
that each of them indicates "just where things are". 
Location is not something supernaturally revealed to the 
mind; it is a kind of conventional summary of those 
properties or relations of objects which condition certain 
visual and tactual sensations. 

Does not this show that "right" location in space 
cannot be nearly so important and fundamental as it is 
made out to be in the Newtonian scheme of things? 
The different observers are able to play fast and loose 
with it without ill effects. 

Suppose that location is, I will not say entirely a 
myth, but not quite the definite thing it is made out to 
be in classical physics; that the Newtonian idea of 
location contains some truth and some padding, and it 
is not the truth but the padding that our observers are 
quarrelling over. That would explain a great deal. It 
would explain, for instance, why all the forces of Nature 
seem to have entered into a conspiracy to prevent our 
discovering the definite location of any object (its posi- 
tion in the "right" frame of space) ; naturally they 
cannot reveal it, if it does not exist. 

This thought will be followed up in the next chapter. 
Meanwhile let us glance back over the arguments that 
have led to the present situation. It arises from the 
failure of our much-trusted measuring scale, a failure 
which we can infer from strong experimental evidence 
or more simply as an inevitable consequence of accepting 
the electrical theory of matter. This unforeseen be- 
haviour is a constant property of all kinds of matter and 
is even shared by optical and electrical measuring devices. 



SUMMARY 19 

Thus it is not betrayed by any kind of discrepancy in 
applying the usual methods of measurement. The dis- 
crepancy is revealed when we change the standard 
motion of the measuring appliances, e.g. when we com- 
pare lengths and distances as measured by terrestrial 
observers with those which would be measured by 
observers on a planet with different velocity. Provision- 
ally we shall call the measured lengths which contain 
this discrepancy "fictitious lengths". 

According to the Newtonian scheme length is definite 
and unique; and each observer should apply corrections 
(dependent on his motion) to reduce his fictitious lengths 
to the unique Newtonian length. But to this there are 
two objections. The corrections to reduce to Newtonian 
length are indeterminate; we know the corrections 
necessary to reduce our own fictitious lengths to those 
measured by an observer with any other prescribed 
motion, but there is no criterion for deciding which 
system is the one intended in the Newtonian scheme. 
Secondly, the whole of present-day physics has been 
based on lengths measured by terrestrial observers 
without this correction, so that whilst its assertions 
ostensibly refer to Newtonian lengths they have actually 
been proved for fictitious lengths. 

The FitzGerald contraction may seem a little thing 
to bring the whole structure of classical physics tumbling 
down. But few indeed are the experiments contributing 
to our scientific knowledge which would not be invali- 
dated if our methods of measuring lengths were funda- 
mentally unsound. We now find that there is no 
guarantee that they are not subject to a systematic kind 
of error. Worse still we do not know if the error 
occurs or not, and there is every reason to presume 
that it is impossible to know. 



Chapter II 

RELATIVITY 

Einstein's Principle. The modest observer mentioned in 
the first chapter was faced with the task of choosing 
between a number of frames of space with nothing to 
guide his choice. They are different in the sense that they 
frame the material objects of the world, including the 
observer himself, differently; but they are indistinguish- 
able in the sense that the world as framed in one space 
conducts itself according to precisely the same laws 
as the world framed in another space. Owing to the 
accident of having been born on a particular planet 
our observer has hitherto unthinkingly adopted one of 
the frames; but he realises that this is no ground for 
obstinately asserting that it must be the right frame. 
Which is the right frame? 

At this juncture Einstein comes forward with a sug- 
gestion — 

"You are seeking a frame of space which you call 
the right frame. In what does its rightness consist?" 

You are standing with a label in your hand before a 
row of packages all precisely similar. You are worried 
because there is nothing to help you decide which of 
the packages it should be attached to. Look at the label 
and see what is written on it. Nothing. 

"Right" as applied to frames of space is a blank label. 
It implies that there is something distinguishing a right 
frame from a wrong frame; but when we ask what is 
this distinguishing property, the only answer we receive 
is "Rightness", which does not make the meaning clearer 
or convince us that there is a meaning. 

20 



EINSTEIN'S PRINCIPLE 21 

I am prepared to admit that frames of space in spite 
of their present resemblance may in the future turn out 
to be not entirely indistinguishable. (I deem it unlikely, 
but I do not exclude it.) The future physicist might 
find that the frame belonging to Arcturus, say, is unique 
as regards some property not yet known to science. Then 
no doubt our friend with the label will hasten to affix 
it. "I told you so. I knew I meant something 
when I talked about a right frame." But it does not 
seem a profitable procedure to make odd noises on 
the off-chance that posterity will find a significance to 
attribute to them. To those who now harp on a right 
frame of space we may reply in the words of Bottom the 
weaver — 

"Who would set his wit to so foolish a bird? Who 
would give a bird the lie, though he cry 'cuckoo' never 
so?" 

And so the position of Einstein's theory is that the 
question of a unique right frame of space does not arise. 
There is a frame of space relative to a terrestrial ob- 
server, another frame relative to the nebular observers, 
others relative to other stars. Frames of space are rela- 
tive. Distances, lengths, volumes — all quantities of 
space-reckoning which belong to the frames — are likewise 
relative. A distance as reckoned by an observer on one 
star is as good as the distance reckoned by an observer 
on another star. We must not expect them to agree; 
the one is a distance relative, to one frame, the other is 
a distance relative to another frame. Absolute distance, 
not relative to some special frame, is meaningless. 

The next point to notice is that the other quantities 
of physics go along with the frame of space, so that they 
also are relative. You may have seen one of those tables 
of "dimensions" of physical quantities showing how 



22 RELATIVITY 

they are all related to the reckoning of length, time and 
mass. If you alter the reckoning of length you alter the 
reckoning of other physical quantities. 

Consider an electrically charged body at rest on the 
earth. Since it is at rest it gives an electric field but no 
magnetic field. But for the nebular physicist it is a 
charged body moving at iooo miles a second. A moving 
charge constitutes an electric current which in accordance 
with the laws of electromagnetism gives rise to a mag- 
netic field. How can the same body both give and 
not give a magnetic field? On the classical theory we 
should have had to explain one of these results as an 
illusion. (There is no difficulty in doing that; only there 
is nothing to indicate which of the two results is the one 
to be explained away.) On the relativity theory both 
results are accepted. Magnetic fields are relative. 
There is no magnetic field relative to the terrestrial 
frame of space; there is a magnetic field relative to 
the nebular frame of space. The nebular physicist will 
duly detect the magnetic field with his instruments 
although our instruments show no magnetic field. That 
is because he uses instruments at rest on his planet and 
we use instruments at rest on ours; or at least we correct 
our observations to accord with the indications of instru- 
ments at rest in our respective frames of space. 

Is there really a magnetic field or not? This is like 
the previous problem of the square and the oblong. 
There is one specification of the field relative to one 
planet, another relative to another. There is no abso- 
lute specification. 

It is not quite true to say that all the physical quan- 
tities are relative to frames of space. We can construct 
new physical quantities by multiplying, dividing, etc.; 
thus we multiply mass and velocity to give momentum, 



RELATIVE AND ABSOLUTE QUANTITIES 23 

divide energy by time to give horse-power. We can set 
ourselves the mathematical problem of constructing in 
this way quantities which shall be invariant, that is to 
say, shall have the same measure whatever frame of 
space may be used. One or two of these invariants 
turn out to be quantities already recognised in pre- 
relativity physics; "action" and "entropy" are the best 
known. Relativity physics is especially interested in 
invariants, and it has discovered and named a few more. 
It is a common mistake to suppose that Einstein's theory 
of relativity asserts that everything is relative. Actually 
it says, "There are absolute things in the world but 
you must look deeply for them. The things that first 
present themselves to your notice are for the most part 
relative." 

Relative and Absolute Quantities. I will try to make 
clear the distinction between absolute and relative quan- 
tities. Number (of discrete individuals) is absolute. It 
is the result of counting, and counting is an absolute 
operation. If two men count the number of people in 
this room and reach different results, one of them must 
be wrong. 

The measurement of distance is not an absolute 
operation. It is possible for two men to measure the 
same distance and reach different results, and yet neither 
of them be wrong. 

I mark two dots on the^ blackboard and ask two 
students to measure very accurately the distance between 
them. In order that there may be no possible doubt as 
to what I mean by distance I give them elaborate 
instructions as to the standard to be used and the pre- 
cautions necessary to obtain an accurate measurement 
of distance. They bring me results which differ. I ask 



24 RELATIVITY 

them to compare notes to find out which of them is 
wrong, and why? Presently they return and say: "It 
was your fault because in one respect your instructions 
were not explicit. You did not mention what motion 
the scale should have when it was being used." One 
of them without thinking much about the matter had 
kept the scale at rest on the earth. The other had 
reflected that the earth was a very insignificant planet of 
which the Professor had a low opinion. He thought it 
would be only reasonable to choose some more impor- 
tant body to regulate the motion of the scale, and so he 
had given it a motion agreeing with that of the enor- 
mous star Betelgeuse. Naturally the FitzGerald contrac- 
tion of the scale accounted for the difference of results. 

I am disinclined to accept this excuse. I say severely, 
"It is all nonsense dragging in the earth or Betel- 
geuse or any other body. You do not require any 
standard external to the problem. I told you to measure 
the distance of two points on the blackboard; you should 
have made the motion of the scale agree with that of 
the blackboard. Surely it is commonsense to make your 
measuring scale move with what you are measuring. 
Remember that next time." 

A few days later I ask them to measure the wave- 
length of sodium light — the distance from crest to crest 
of the light waves. They do so and return in triumphal 
agreement: ''The wave-length is infinite". I point out 
to them that this does not agree with the result given 
in the book (.000059 cm.). "Yes", they reply, u we 
noticed that; but the man in the book did not do it 
right. You told us always to make the measuring scale 
move with the thing to be measured. So at great trouble 
and expense we sent our scales hurtling through the 
laboratory at the same speed as the light." At this speed 



RELATIVE AND ABSOLUTE QUANTITIES 25 

the FitzGerald contraction is infinite, the metre rods 
contract to nothing, and so it takes an infinite number 
of them to fill up the interval from crest to crest of the 
waves. 

My supplementary rule was in a way quite a good 
rule; it would always give something absolute — some- 
thing on which they would necessarily agree. Only 
unfortunately it would not give the length or distance. 
When we ask whether distance is absolute or relative, 
we must not first make up our minds that it ought to 
be absolute and then change the current significance of 
the term to make it so. 

Nor can we altogether blame our predecessors for 
having stupidly made the word "distance" mean some- 
thing relative when they might have applied it to a 
result of spatial measurement which was absolute and 
unambiguous. The suggested supplementary rule has 
one drawback. We often have to consider a system 
containing a number of bodies with different motions; 
it would be inconvenient to have to measure each body 
with apparatus in a different state of motion, and we 
should get into a terrible muddle in trying to fit the 
different measures together. Our predecessors were 
wise in referring all distances to a single frame of space, 
even though their expectation that such distances would 
be absolute has not been fulfilled. 

As for the absolute quantity given by the proposed 
supplementary rule, we may set it alongside distances 
relative to the earth and distances relative to Betelgeuse, 
etc., as a quantity of some interest to study. It is called 
"proper-distance". Perhaps you feel a relief at getting 
hold of something absolute and would wish to follow 
it up. Excellent. But remember this will lead you away 
from the classical scheme of physics which has chosen 



26 RELATIVITY 

the relative distances to build on. The quest of the 
absolute leads into the four-dimensional world. 

A more familiar example of a relative quantity is 
"direction" of an object. There is a direction of Cam- 
bridge relative to Edinburgh and another direction rela- 
tive to London, and so on. It never occurs to us to 
think of this as a discrepancy, or to suppose that there 
must be some direction of Cambridge (at present undis- 
coverable) which is absolute. The idea that there ought 
to be an absolute distance between two points contains 
the same kind of fallacy. There is, of course, a differ- 
ence of detail; the relative direction above mentioned is 
relative to a particular position of the observer, whereas 
the relative distance is relative to a particular velocity 
of the observer. We can change position freely and 
so introduce large changes of relative direction; but 
we cannot change velocity appreciably — the 300 miles 
an hour attainable by our fastest devices being too 
insignificant to count. Consequently the relativity of 
distance is not a matter of common experience as the 
relativity of direction is. That is why we have unfor- 
tunately a rooted impression in our minds that distance 
ought to be absolute. 

A very homely illustration of a relative quantity is 
afforded by the pound sterling. Whatever may have 
been the correct theoretical view, the man in the street 
until very recently regarded a pound as an absolute 
amount of wealth. But dire experience has now con- 
vinced us all of its relativity. At first we used to cling 
to the idea that there ought to be an absolute pound 
and struggle to express the situation in paradoxical state- 
ments — the pound had really become seven-and-six- 
pence. But we have grown accustomed to the situation 
and continue to reckon wealth in pounds as before, 



NATURE'S PLAN OF STRUCTURE 27 

merely recognising that the pound is relative and there- 
fore must not be expected to have those properties that 
we had attributed to it in the belief that it was absolute. 
You can form some idea of the essential difference in 
the outlook of physics before and after Einstein's 
principle of relativity by comparing it with the difference 
in economic theory which comes from recognising the 
•relativity of value of money. I suppose that in stable 
times the practical consequences of this relativity are 
manifested chiefly in the minute fluctuations of foreign 
exchanges, which may be compared with the minute 
changes of length affecting delicate experiments like the 
Michelson-Morley experiment. Occasionally the con- 
sequences may be more sensational — a mark-exchange 
soaring to billions, a high-speed 8 particle contracting 
to a third of its radius. But it is not these casual mani- 
festations which are the main outcome. Clearly an 
economist who believes in the absoluteness of the pound 
has not grasped the rudiments of his subject. Similarly 
if we have conceived the physical world as intrinsically 
constituted out of those distances, forces and masses 
which are now seen to have reference only to our own 
special reference frame, we are far from a proper under- 
standing of the nature of things. 

Nature's Plan of Structure. Let us now return to the 
observer who was so anxious to pick out a "right" 
frame of space. I suppose that what he had in mind 
was to find Nature's own frame — the frame on which 
Nature based her calculations when she poised the 
planets under the law of gravity, or the reckoning of 
symmetry which she used when she turned the electrons 
on her lathe. But Nature has been too subtle for him; 
she has not left anything to betray the frame which she 



28 RELATIVITY 

used. Or perhaps the concealment is not any particular 
subtlety; she may have done her work without employing 
a frame of space. Let me tell you a parable. 

There was once an archaeologist who used to com- 
pute the dates of ancient temples from their orientation. 
He found that they were aligned with respect to the 
rising of particular stars. Owing to precession the 
star no longer rises in the original line, but the date 
when it was rising in the line of the temple can be 
calculated, and hence the epoch of construction of the 
temple is discovered. But there was one tribe for 
which this method would not work; they had built only 
circular temples. To the archaeologist this seemed a 
manifestation of extraordinary subtlety on their part; 
they had hit on a device which would conceal entirely 
the date when their temples were constructed. One 
critic, however, made the ribald suggestion that per- 
haps this particular tribe was not enthusiastic about 
astronomy. 

Like the critic I do not think Nature has been par- 
ticularly subtle in concealing which frame she prefers. 
It is just that she is not enthusiastic about frames of 
space. They are a method of partition which w T e have 
found useful for reckoning, but they play no part in 
the architecture of the universe. Surely it is absurd to 
suppose that the universe is planned in such a way as to 
conceal its plan. It is like the schemes of the White 
Knight — 

But I was thinking of a plan 

To dye one's whiskers green, 

And always use so large a fan 

That they could not be seen. 

If this is so we shall have to sweep away the frames 
of space before we can see Nature's plan in its real 



NATURE'S PLAN OF STRUCTURE 29 

significance. She herself has paid no attention to them, 
and they can only obscure the simplicity of her scheme. 
I do not mean to suggest that we should entirely rewrite 
physics, eliminating all reference to frames of space or 
any quantities referred to them; science has many tasks 
to perform, besides that of apprehending the ultimate 
plan of structure of the world. But if we do wish to 
have insight on this latter point, then the first step is to 
make an escape from the irrelevant space-frames. 

This will involve a great change from classical con- 
ceptions, and important developments will follow from 
our change of attitude. For example, it is known that 
both gravitation and electric force follow approximately 
the law of inverse-square of the distance. This law 
appeals strongly to us by its simplicity; not only is it 
mathematically simple but it corresponds very naturally 
with the weakening of an effect by spreading out in 
three dimensions. We suspect therefore that it is 
likely to be the exact law of gravitational and electric 
fields. But although it is simple for us it is far from 
simple for Nature. Distance refers to a space-frame; 
it is different according to the frame chosen. We cannot 
make sense of the law of inverse-square of the distance 
unless we have first fixed on a frame of space; but 
Nature has not fixed on any one frame. Even if by 
some self-compensation the law worked out so as to give 
the same observable consequences whatever space-frame 
we might happen to choose (which it does not) we should 
still be misapprehending its real mode of operation. In 
chapter VI we shall try to gain a new insight into the 
law (which for most practical applications is so nearly 
expressed by the inverse-square) and obtain a picture 
of its working which does not drag in an irrelevant frame 
of space. The recognition of relativity leads us to 



30 RELATIVITY 

seek a new way of unravelling the complexity of natural 
phenomena. 

Velocity through the Aether. The theory of relativity is 
evidently bound up with the impossibility of detecting 
absolute velocity; if in our quarrel with the nebular 
physicists one of us had been able to claim to be 
absolutely at rest, that would be sufficient reason for 
preferring the corresponding frame. This has some- 
thing in common with the well-known philosophic belief 
that motion must necessarily be relative. Motion is 
change of position relative to something-, if we try to 
think of change of position relative to nothing the whole 
conception fades away. But this does not completely 
settle the physical problem. In physics we should not 
be quite so scrupulous as to the use of the word absolute. 
Motion with respect to aether or to any universally sig- 
nificant frame would be called absolute. 

No aethereal frame has been found. We can only 
discover motion relative to the material landmarks 
scattered casually about the world; motion with respect 
to the universal ocean of aether eludes us. We say, 
"Let V be the velocity of a body through the aether", 
and form the various electromagnetic equations in which 
V is scattered liberally. Then we insert the observed 
values, and try to eliminate everything that is unknown 
except V. The solution goes on famously; but just as 
we have got rid of the other unknowns, behold! V dis- 
appears as well, and we are left with the indisputable 
but irritating conclusion — 

This is a favourite device that mathematical equations 
resort to, when we propound stupid questions. If we 
tried to find the latitude and longitude of a point north- 



VELOCITY THROUGH THE AETHER 31 

east from the north pole we should probably receive 
the same mathematical answer. "Velocity through 
aether" is as meaningless as "north-east from the north 
pole". 

This does not mean that the aether is abolished. We 
need an aether. The physical world is not to be analysed 
into isolated particles of matter or electricity with 
featureless interspace. We have to attribute as much 
character to the interspace as to the particles, and in 
present-day physics quite an army of symbols is required 
to describe what is going on in the interspace. We 
postulate aether to bear the characters of the interspace 
as we postulate matter or electricity to bear the charac- 
ters of the particles. Perhaps a philosopher might ques- 
tion whether it is not possible to admit the characters 
alone without picturing anything to support them — thus 
doing away with aether and matter at one stroke. But 
that is rather beside the point. 

In the last century it was widely believed that aether 
was a kind of matter, having properties such as mass, 
rigidity, motion, like ordinary matter. It would be 
difficult to say when this view died out. It probably 
lingered longer in England than on the continent, but 
I think that even here it had ceased to be the orthodox 
view some years before the advent of the relativity 
theory. Logically it was abandoned by the numerous 
nineteenth-century investigators who regarded matter 
as vortices, knots, squirts, etc., in the aether; for clearly 
they could not have supposed that aether consisted of 
vortices in the aether. But it may not be safe to assume 
that the authorities in question were logical. 

Nowadays it is agreed that aether is not a kind of 
matter. Being non-material, its properties are sui generis. 
We must determine them by experiment; and since we 



32 RELATIVITY 

have no ground for any preconception, the experimental 
conclusions can be accepted without surprise or mis- 
giving. Characters such as mass and rigidity which we 
meet with in matter will naturally be absent in aether; 
but the aether will have new and definite characters of 
its own. In a material ocean we can say that a particu- 
lar particle of water which was here a few moments ago 
is now over there; there is no corresponding assertion 
that can be made about the aether. If you have been 
thinking of the aether in a way which takes for granted 
this property of permanent identification of its particles, 
you must revise your conception in accordance with the 
modern evidence. We cannot find our velocity through 
the aether; we cannot say whether the aether now in this 
room is flowing out through the north wall or the south 
wall. The question would have a meaning for a mate- 
rial ocean, but there is no reason to expect it to have a 
meaning for the non-material ocean of aether. 

The aether itself is as much to the fore as ever it was, 
in our present scheme of the world. But velocity through 
aether has been found to resemble that elusive lady 
Mrs. Harris; and Einstein has inspired us with the 
daring scepticism — "I don't believe there's no sich a 
person". 

Is the FitzGerald Contraction Real? I am often asked 
whether the FitzGerald contraction really occurs. It 
was introduced in the first chapter before the idea of 
relativity was mentioned, and perhaps it is not quite 
clear what has become of it now that the theory of 
relativity has given us a new conception of what is going 
on in the world. Naturally my first chapter, which 
describes the phenomena according to the ideas of 
classical physics in order to show the need for a new 



IS FITZGERALD CONTRACTION REAL? 33 

theory, contains many statements which we should 
express differently in relativity physics. 

Is it really true that a moving rod becomes shortened 
in the direction of its motion? It is not altogether easy 
to give a plain answer. I think we often draw a dis- 
tinction between what is true and what is really true. A 
statement which does not profess to deal with anything 
except appearances may be true; a statement which is 
not only true but deals with the realities beneath the 
appearances is really true. 

You receive a balance-sheet from a public company 
and observe that the assets amount to such and such a 
figure. Is this true? Certainly; it is certified by a 
chartered accountant. But is it really true? Many 
questions arise; the real values of items are often very 
different from those which figure in the balance-sheet. 
I am not especially referring to fraudulent companies. 
There is a blessed phrase "hidden reserves"; and gen- 
erally speaking the more respectable the company the 
more widely does its balance-sheet deviate from reality. 
This is called sound finance. But apart from deliberate 
use of the balance-sheet to conceal the actual situation, 
it is not well adapted for exhibiting realities, because 
the main function of a balance-sheet is to balance 
and everything else has to be subordinated to that 
end. 

The physicist who uses a frame of space has to 
account for every millimetre of space — in fact to draw 
up a balance-sheet, and make it balance. Usually there 
is not much difficulty. But suppose that he happens to 
be concerned with a man travelling at 161^000 miles 
a second. The man is an ordinary 6-foot man. So far 
as reality is concerned the proper entry in the balance- 
sheet would appear to be 6 feet. But then the balance- 



34 RELATIVITY 

sheet would not balance. In accounting for the rest of 
space there is left only 3 feet between the crown of his 
head and the soles of his boots. His balance-sheet 
length is therefore "written down" to 3 feet. 

The writing-down of lengths for balance-sheet pur- 
poses is the FitzGerald contraction. The shortening of 
the moving rod is true, but it is not really true. It is not 
a statement about reality (the absolute) but it is a true 
statement about appearances in our frame of reference.* 
An object has different lengths in the different space- 
frames, and any 6-foot man will have a length 3 feet in 
some frame or other. The statement that the length of 
the rapid traveller is 3 feet is true, but it does not indicate 
any special peculiarity about the man; it only indicates 
that our adopted frame is the one in which his length is 
3 feet. If it hadn't been ours, it would have been some- 
one else's. 

Perhaps you will think we ought to alter our method 
of keeping the accounts of space so as to make them 
directly represent the realities. That would be going to 
a lot of trouble to provide for what are after all rather 
rare transactions. But as a matter of fact we have 
managed to meet your desire. Thanks to Minkowski 
a way of keeping accounts has been found which 
exhibits realities (absolute things) and balances. There 
has been no great rush to adopt it for ordinary purposes 
because it is a four-dimensional balance-sheet. 

Let us take a last glance back before we plunge into 

*The proper-length (p. 25) is unaltered; but the relative length is 
shortened. We have already seen that the word "length" as currently- 
used refers to relative length, and in confirming the statement that the 
moving rod changes its length we are, of course, assuming that the word 
is used with its current meaning. 



SUMMARY 35 

four dimensions. We have been confronted with some- 
thing not contemplated in classical physics — a multi- 
plicity of frames of space, each one as good as any 
other. And in place of a distance, magnetic force, 
acceleration, etc., which according to classical ideas 
must necessarily be definite and unique, we are con- 
fronted with different distances, etc., corresponding to 
the different frames, with no ground for making a choice 
between them. Our simple solution has been to give 
up the idea that one of these is right and that the others 
are spurious imitations, and to accept them en bloc; so 
that distance, magnetic force, acceleration, etc., are 
relative quantities, comparable with other relative quan- 
tities already known to us such as direction or velocity. 
In the main this leaves the structure of our physical 
knowledge unaltered; only we must give up certain 
expectations as to the behaviour of these quantities, and 
certain tacit assumptions which were based on the belief 
that they are absolute. In particular a law of Nature 
which seemed simple and appropriate for absolute quan- 
tities may be quite inapplicable to relative quantities and 
therefore require some tinkering. Whilst the structure of 
our physical knowledge is not much affected, the change 
in the underlying conceptions is radical. We have trav- 
elled far from the old standpoint which demanded 
mechanical models of everything in Nature, seeing that 
we do not now admit even a definite unique distance 
between two points. The relativity of the current scheme 
of physics invites us to search deeper and find the abso- 
lute scheme underlying it, so that we may see the world 
in a truer perspective. 



Chapter III 

TIME 

Astronomer Royal's Time. I have sometimes thought it 
would be very entertaining to hear a discussion between 
the Astronomer Royal and, let us say, Prof. Bergson on 
the nature of time. Prof. Bergson's authority on the 
subject is well known; and I may remind you that the 
Astronomer Royal is entrusted with the duty of finding 
out time for our everyday use, so presumably he has 
some idea of what he has to find. I must date the 
discussion some twenty years back, before the spread of 
Einstein's ideas brought about a rapprochement. There 
would then probably have been a keen disagreement, 
and I rather think that the philosopher would have had 
the best of the verbal argument. After showing that 
the Astronomer Royal's idea of time was quite non- 
sensical, Prof. Bergson would probably end the dis- 
cussion by looking at his watch and rushing off to catch 
a train which was starting by the Astronomer Royal's 
time. 

Whatever may be time de ]ure } the Astronomer 
Royal's time is time de facto. His time permeates every 
corner of physics. It stands in no need of logical de- 
fence; it is in the much stronger position of a vested 
interest. It has been woven into the structure of the 
classical physical scheme. "Time" in physics means 
Astronomer Royal's time. You may be aware that it is 
revealed to us in Einstein's theory that time and space 
are mixed up in a rather strange way. This is a great 
stumbling-block to the beginner. He is inclined to say, 
"That is impossible. I feel it in my bones that time and 

36 



ASTRONOMER ROYAL'S TIME 37 

space must be of entirely different nature. They cannot 
possibly be mixed up." The Astronomer Royal com- 
placently retorts, "It is not impossible. / have mixed 
them up." Well, that settles it. If the Astronomer 
Royal has mixed them, then his mixture will be the 
groundwork of present-day physics. 

We have to distinguish two questions which are not 
necessarily identical. First, what is the true nature of 
time? Second, what is the nature of that quantity which 
has under the name of time become a fundamental part 
of the structure of classical physics? By long history 
of experiment and theory the results of physical inves- 
tigation have been woven into a scheme which has on 
the whole proved wonderfully successful. Time — the 
Astronomer Royal's time — has its importance from the 
fact that it is a constituent of that scheme, the binding 
material or mortar of it. That importance is not les- 
sened if it should prove to be only imperfectly repre- 
sentative of the time familiar to our consciousness. We 
therefore give priority to the second question. 

But I may add that Einstein's theory, having cleared 
up the second question, having found that physical 
time is incongruously mixed with space, is able to pass 
on to the first question. There is a quantity, unrecog- 
nised in pre-relativity physics, which more directly 
represents the time known to consciousness. This is 
called proper-time or interval. It is definitely separate 
from and unlike proper-space. Your protest in the 
name of commonsense against a mixing of time and 
space is a feeling which I desire to encourage. Time and 
space ought to be separated. The current representa- 
tion of the enduring world as a three-dimensional space 
leaping from instant to instant through time is an 
unsuccessful attempt to separate them. Come back with 



38 TIME 

me into the virginal four-dimensional world and we will 
carve it anew on a plan which keeps them entirely 
distinct. We can then resurrect the almost forgotten 
time of consciousness and find that it has a gratifying 
importance in the absolute scheme of Nature. 

But first let us try to understand why physical time 
has come to deviate from time as immediately perceived. 
We have jumped to certain conclusions about time and 
have come to regard them almost as axiomatic, although 
they are not really justified by anything in our immediate 
perception of time. Here is one of them. 

If two people meet twice they must have lived the 
same time between the two meetings, even if one of 
them has travelled to a distant part of the universe and 
back in the interim. 

An absurdly impossible experiment, you will say. 
Quite so; it is outside all experience. Therefore, may 
I suggest that you are not appealing to your experience 
of time when you object to a theory which denies the 
above statement? And yet if the question is pressed 
most people would answer impatiently that of course 
the statement is true. They have formed a notion of 
time rolling on outside us in a way which makes this 
seem inevitable. They do not ask themselves whether 
this conclusion is warranted by anything in their actual 
experience of time. 

Although we cannot try the experiment of sending a 
man to another part of the universe, we have enough 
scientific knowledge to compute the rates of atomic and 
other physical processes in a body at rest and a body 
travelling rapidly. We can say definitely that the bodily 
processes in the traveller occur more slowly than the 
corresponding processes in the man at rest (i.e. more 
slowly according to the Astronomer Royal's time). This 






ASTRONOMER ROYAL'S TIME 39 

is not particularly mysterious; it is well known both 
from theory and experiment that the mass or inertia of 
matter increases when the velocity increases. The re- 
tardation is a natural consequence of the greater inertia. 
Thus so far as bodily processes are concerned the fast- 
moving traveller lives more slowly. His cycle of diges- 
tion and fatigue; the rate of muscular response to stim- 
ulus; the development of his body from youth to age; 
the material processes in his brain which must more or 
less keep step with the passage of thoughts and emo- 
tions; the watch which ticks in his waistcoat pocket; all 
these must be slowed down in the same ratio. If the 
speed of travel is very great we may find that, whilst 
the stay-at-home individual has aged 70 years, the trav- 
eller has aged 1 year. He has only found appetite for 
365 breakfasts, lunches, etc.; his intellect, clogged by a 
slow-moving brain, has only traversed the amount of 
thought appropriate to one year of terrestrial life. His 
watch, which gives a more accurate and scientific reck- 
oning, confirms this. Judging by the time which con- 
sciousness attempts to measure after its own rough 
fashion — and, I repeat, this is the only reckoning of time 
which we have a right to expect to be distinct from 
space — the two men have not lived the same time 
between the two meetings. 

Reference to time as estimated by consciousness is 
complicated by the fact that the reckoning is very erratic. 
"I'D tell you who Time ambles withal, who Time trots 
withal, who Time gallops withal, and who he stands 
still withal." I have not been referring to these sub- 
jective variations. I do not very willingly drag in 
so unsatisfactory a time-keeper; only I have to deal 
with the critic who tells me what "he feels in his bones" 
about time, and I would point out to him that the basis 



40 TIME 

of that feeling is time lived, which we have just seen 
may be 70 years for one individual and 1 year for 
another between their two meetings. We can reckon 
"time lived' 5 quite scientifically, e.g. by a watch travel- 
ling with the individual concerned and sharing his 
changes of inertia with velocity. But there are obvious 
drawbacks to the general adoption of "time lived". It 
might be useful for each individual to have a private 
time exactly proportioned to his time lived; but it would 
be extremely inconvenient for making appointments. 
Therefore the Astronomer Royal has adopted a uni- 
versal time-reckoning which does not follow at all strictly 
the time lived. According to it the time-lapse does not 
depend on how the object under consideration has 
moved in the meanwhile. I admit that this reckoning 
is a little hard on our returned traveller, who will be 
counted by it as an octogenarian although he is to all 
appearances still a boy in his teens. But sacrifices must 
be made for the general benefit. In practice we have 
not to deal with human beings travelling at any great 
speed; but we have to deal with atoms and electrons 
travelling at terrific speed, so that the question of pri- 
vate time-reckoning versus general time-reckoning is a 
very practical one. 

Thus in physical time (or Astronomer Royal's time) 
two people are deemed to have lived the same time 
between two meetings, whether or not that accords with 
their actual experience. The consequent deviation from 
the time of experience is responsible for the mixing 
up of time and space, which, of course, would be 
impossible if the time of direct experience had been 
rigidly adhered to. Physical time is, like space, a kind 
of frame in which we locate the events of the external 
world. We are now going to consider how in practice 



LOCATION OF EVENTS 



4i 



external events are located in a frame of space and time. 
We have seen that there is an infinite choice of alter- 
native frames; so, to be quite explicit, I will tell you 
how / locate events in my frame. 

Location of Events, In Fig. 1 you see a collection of 
events, indicated by circles. They are not at present in 









FUTURE 
O 








u 












Ld 


cc 












q: 


LxJ 


O 




©HERE-NOW 


O 


O 


Ld 


I 












X 


Ld 













Ld 


if> 




O 








<o 


-J 




^m^ 

^+^ 








-J 


Ld 




O 


°o 






Ld 


. 










O 










PAST 









Fig. 1 

their right places; that is the job before me — to put 
them into proper location in my frame of space and 
time. Among them I can immediately recognise and 
label the event Here-Now, viz. that which is happening 
in this room at this moment. The other events are at 
varying degrees of remoteness from Here-Now, and it 



42 TIME 

is obvious to me that the remoteness is not only of 
different degrees but of different kinds. Some events 
spread away towards what in a general way I call the 
Past; I can contemplate others which are distant in 
the Future; others are remote in another kind of way 
towards China or Peru, or in general terms Elsewhere. 
In this picture I have only room for one dimension of 
Elsewhere; another dimension sticks out at right angles 
to the paper; and you must imagine the third dimension 
as best you can. 

Now we must pass from this vague scheme of location 
to a precise scheme. The first and most important thing 
is to put Myself into the picture. It sounds egotistical; 
but, you see, it is my frame of space that will be used, 
so it all hangs round me. Here I am — a kind of four- 
dimensional worm (Fig. 2). It is a correct portrait; 
I have considerable extension towards the Past and 
presumably towards the Future, and only a moderate 
extension towards Elsewhere. The "instantaneous me", 
i.e. myself at this instant, coincides with the event Here- 
Now. Surveying the world from Here-Now, I can see 
many other events happening now. That puts it into my 
head that the instant of which I am conscious here must 
be extended to include them; and I jump to the con- 
clusion that Now is not confined to Here-Now. I there- 
fore draw the instant Now, running as a clean section 
across the world of events, in order to accommodate all 
the distant events which are happening now. I select 
the events which I see happening now and place them 
on this section, which I call a moment of time or an 
"instantaneous state of the world". I locate them on 
Now because they seem to be Now. 

This method of location lasted until the year 1667, 
when it was found impossible to make it work consist- 



LOCATION OF EVENTS 



43 



ently. It was then discovered by the astronomer Roemer 
that what is seen now cannot be placed on the instant 
Now. (In ordinary parlance — light takes time to travel.) 
That was really a blow to the whole system of world- 
wide instants, which were specially invented to accommo- 
date these events. We had been mixing up two distinct 





FUT 


URE 


Id 




Ld 


^ NOW 


c 


N HERE-NOW /-., NOW *^ 


Ld " ' "" 


^A 


A ^ '" K..' " Ld 


1 ^*'*' 

id-**- 


"* Id 


</> 




<n 


_i 




-j 


Ld 


u. 

_J 
111 

(0 

> 
5 


id 




PAST 



Fig. 2^ 

events; there was the original event somewhere out in 
the external world and there was a second event, viz. 
the seeing by us of the first event. The second event 
was in our bodies Here-Now; the first event was neither 
Here nor Now. The experience accordingly gives no 
indication of a Now which is not Here; and we might 



44 TIME 

well have abandoned the idea that we have intuitive 
recognition of a Now other than Here-Now, which was 
the original reason for postulating world-wide instants 
Now. 

However, having become accustomed to world-wide 
instants, physicists were not ready to abandon them. 
And, indeed, they have considerable usefulness pro- 
vided that we do not take them too seriously. They were 
left in as a feature of the picture, and two Seen-Now 
lines were drawn, sloping backwards from the Now line, 
on which events seen now could be consistently placed. 
The cotangent of the angle between the Seen-Now lines 
and the Now line was interpreted as the velocity of light. 

Accordingly when I see an event in a distant part of 
the universe, e.g. the outbreak of a new star, I locate it 
(quite properly) on the Seen-Now line. Then I make a 
certain calculation from the measured parallax of the 
star and draw my Now line to pass, say, 300 years in 
front of the event, and my Now line of 300 years ago 
to pass through the event. By this method I trace the 
course of my Now lines or world-wide instants among 
the events, and obtain a frame of time-location for 
external events. The auxiliary Seen-Now lines, having 
served their purpose, are rubbed out of the picture. 

That is how / locate events; how about youf We 
must first put You into the picture (Fig. 3). We shall 
suppose that you are on another star moving with 
different velocity but passing close to the earth at the 
present moment. You and I were far apart in the past 
and will be again in the future, but we are both Here- 
Now. That is duly shown in the picture. We survey 
the world from Here-Now, and of course we both see 
the same events simultaneously. We may receive rather 
different impressions of them; our different motions 



LOCATION OF EVENTS 



45 



will cause different Doppler effects, FitzGerald con- 
tractions, etc. There may be slight misunderstandings 
until we realise that what you describe as a red square 
is what I would describe as a green oblong, and so on. 
But, allowing for this kind of difference of description, 



FUTURE 



id 

cc MY NQW 

Id Y oUR N°^ 




Id 




HERE-NOW 



v0 ornow. 

MY NOW 



**- s £V 



2f? 



W 



Id 

cr 

Ld 

I 

U 

<0 

_J 

Id 



PAST 



Fig. 3 

it will soon become clear that we are looking at the same 
events, and we shall agree entirely as to how the Seen- 
Now lines lie with respect to the events. Starting from 
our common Seen-Now lines, you have next to make the 
calculations for drawing your Now line among the 
events, and you trace it as shown in Fig. 3. 



46 TIME 

How is it that, starting from the same Seen-Now 
lines, you do not reproduce my Now line? It is because 
a certain measured quantity, viz. the velocity of light, 
has to be employed in the calculations; and naturally 
you trust to your measures of it as I trust to mine. 
Since our instruments are affected by different Fitz- 
Gerald contractions, etc., there is plenty of room for 
divergence. Most surprisingly we both find the same 
velocity of light, 299,796 kilometres per second. But 
this apparent agreement is really a disagreement; be- 
cause you take this to be the velocity relative to your 
planet and I take it to be the velocity relative to mine.* 
Therefore our calculations are not in accord, and your 
Now line differs from mine. 

If we believe our world-wide instants or Now lines 
to be something inherent in the world outside us, we 
shall quarrel frightfully. To my mind it is ridiculous 
that you should take events on the right of the picture 
which have not -happened yet and events on the left 
which are already past and call the combination an 
instantaneous condition of the universe. You are 
equally scornful of my grouping. We can never agree. 
Certainly it looks from the picture as though my 
instants were more natural than yours; but that is 
because / drew the picture. You, of course, would 
redraw it with your Now lines at right angles to your- 
self. 



* The measured velocity of light is the average to-and-fro velocity. 
The velocity in one direction singly cannot be measured until after the 
Now lines have been laid down and therefore cannot be used in laying 
down the Now lines. Thus there is a deadlock in drawing the Now lines 
which can only be removed by an arbitrary assumption or convention. 
The convention actually adopted is that (relative to the observer) the 
velocities of light in the two opposite directions are equal. The resulting 
Now lines must therefore be regarded as equally conventional. 



ABSOLUTE PAST AND FUTURE 47 

But we need not quarrel if the Now lines are merely 
reference lines drawn across the world for convenience 
in locating events — like the lines of latitude and longi- 
tude on the earth. There is then no question of a right 
way and a wrong way of drawing the lines; we draw 
them as best suits our convenience. World-wide instants 
are not natural cleavage planes of time; there is nothing 
equivalent to them in the absolute structure of the world; 
they are imaginary partitions which we find it con- 
venient to adopt. 

We have been accustomed to regard the world — the 
enduring world — as stratified into a succession of in- 
stantaneous states. But an observer on another star 
would make the strata run in a different direction from 
ours. We shall see more clearly the real mechanism of 
the physical world if we can rid our minds of this 
illusion of stratification. The world that then stands 
revealed, though strangely unfamiliar, is actually much 
simpler. There is a difference between simplicity and 
familiarity. A pig may be most familiar to us in the 
form of rashers, but the unstratified pig is a simpler 
object to the biologist who wishes to understand how 
the animal functions. 

Absolute Past and Future. Let us now try to attain this 
absolute view. We rub out all the Now lines. We rub 
out Yourself and Myself, since we are no longer 
essential to the world. But the Seen-Now lines are left. 
They are absolute, since all observers from Here-Now 
agree about them. The flat picture is a section; you 
must imagine it rotated (twice rotated in fact, since 
there are two more dimensions outside the picture). The 
Seen-Now locus is thus really a cone ; or by taking account 
of the prolongation of the lines into the future a double 



48 TIME 

cone or hour-glass figure (Fig. 4). These hour-glasses 
(drawn through each point of the world considered in 
turn as a Here-Now) embody what we know of the abso- 
lute structure of the world so far as space and time are 
concerned. They show how the "grain" of the world 
runs. 

Father Time has been pictured as an old man with 
a scythe and an hour-glass. We no longer permit him 
to mow instants through the world with his scythe; but 
we leave him his hour-glass. 



ABSOLUTE FUTURE 



ABSOLUTE ^**^rv^^ ABSOLUTE 

ELSEWHERE ^^^itMtrnvw ELSEWHERE 




ABSOLUTE PAST 



«*g£^~- — -^W 



Fig. 4 

Since the hour-glass is absolute its two cones provide 
respectively an Absolute Future and an Absolute Past 
for the event Here-Now. They are separated by a 
wedge-shaped neutral zone which (absolutely) is neither 
past nor future. The common impression that relativity 
turns past and future altogether topsy-turvy is quite 
false. But, unlike the relative past and future, the 
absolute past and future are not separated by an in- 
finitely narrow present. It suggests itself that the 



ABSOLUTE PAST AND FUTURE 49 

neutral wedge might be called the Absolute Present ; but I 
do not think that is a good nomenclature. It is much 
better described as Absolute Elsewhere. We have 
abolished the Now lines, and in the absolute world the 
present (Now) is restricted to Here-Now. 

Perhaps I may illustrate the peculiar conditions 
arising from the wedge-shaped neutral zone by a rather 
hypothetical example. Suppose that you are in love with 
a lady on Neptune and that she returns the sentiment. 
It will be some consolation for the melancholy separation 
if you can say to yourself at some — possibly pre- 
arranged — moment, "She is thinking of me now". 
Unfortunately a difficulty has arisen because we have 
had to abolish Now. There is no absolute Now, but 
only the various relative Nows differing according to 
the reckoning of different observers and covering the 
whole neutral wedge which at the distance of Neptune 
is about eight hours thick. She will have to think of 
you continuously for eight hours on end in order to 
circumvent the ambiguity of "Now". 

At the greatest possible separation on the earth the 
thickness of the neutral wedge is no more than a tenth 
of a second; so that terrestrial synchronism is not 
seriously interfered with. This suggests a qualification 
of our previous conclusion that the absolute present is 
confined to Here-Now. It is true as regards instan- 
taneous events (point-events). But in practice the 
events we notice are of more than infinitesimal duration. 
If the duration is sufficient to cover the width of the 
neutral zone, then the event taken as a whole may fairly 
be considered to be Now absolutely. From this point 
of view the "nowness" of an event is like a shadow cast 
by it into space, and the longer the event the farther 
will the umbra of the shadow extend. 



50 TIME 

As the speed of matter approaches the speed of light 
its mass increases to infinity, and therefore it is impos- 
sible to make matter travel faster than light. This 
conclusion is deduced from the classical laws of physics, 
and the increase of mass has been verified by experiment 
up to very high velocities. In the absolute world this 
means that a particle of matter can only proceed from 
Here-Now into the absolute future — which, you will 
agree, is a reasonable and proper restriction. It cannot 
travel into the neutral zone; the limiting cone is the 
track of light or of anything moving with the speed of 
light. We ourselves are attached to material bodies, and 
therefore we can only go on into the absolute future. 

Events in the absolute future are not absolutely 
Elsewhere. It would be possible for an observer to 
travel from Here-Now to the event in question in time 
to experience it, since the required velocity is less than 
that of light; relative to the frame of such an observer 
the event would be Here. No observer can reach an 
event in the neutral zone, since the required speed is too 
great. The event is not Here for any observer (from 
Here-Now) ; therefore it is absolutely Elsewhere. 

The Absolute Distinction of Space and Time. By divid- 
ing the world into Absolute Past and Future on the one 
hand and Absolute Elsewhere on the other hand, our 
hour-glasses have restored a fundamental differentiation 
between time and space. It is not a distinction between 
time and space as they appear in a space-time frame, but 
a distinction between temporal and spatial relations. 
Events can stand to us in a temporal relation (absolutely 
past or future) or a spatial relation (absolutely else- 
where), but not in both. The temporal relations radiate 
into the past and future cones and the spatial relations 



DISTINCTION OF SPACE AND TIME 51 

into the neutral wedge; they are kept absolutely sepa- 
rated by the Seen-Now lines which we have identified with 
the grain of absolute structure in the world. We have 
recovered the distinction which the Astronomer Royal 
confused when he associated time with the merely arti- 
ficial Now lines. 

I would direct your attention to an important differ- 
ence in our apprehension of time-extension and space- 
extension. As already explained our course through the 
world is into the absolute future, i.e. along a sequence 
of time-relations. We can never have a similar experi- 
ence of a sequence of space-relations because that 
would involve travelling with velocity greater than light. 
Thus we have immediate experience of the time-relation 
but not of the space-relation. Our knowledge of space- 
relations is indirect, like nearly all our knowledge of the 
external world — a matter of inference and interpretation 
of the impressions which reach us through our sense- 
organs. We have similar indirect knowledge of the 
time-relations existing between the events in the world 
outside us; but in addition we have direct experience 
of the time-relations that we ourselves are traversing — 
a knowledge of time not coming through external sense- 
organs, but taking a short cut into our consciousness. 
When I close my eyes and retreat into my inner mind, 
I feel myself enduring, I do not feel myself extensive. It 
is this feeling of time as affecting ourselves and not 
merely as existing in the relations of external events 
which is so peculiarly characteristic of it; space on the 
other hand is always appreciated as something external. 

That is why time seems to us so much more mysteri- 
ous than space. We know nothing about the intrinsic 
nature of space, and so it is quite easy to conceive it 
satisfactorily. We have intimate acquaintance with the 



52 TIME 

nature of time and so it baffles our comprehension. It 
is the same paradox which makes us believe we under- 
stand the nature of an ordinary table whereas the nature 
of human personality is altogether mysterious. We 
never have that intimate contact with space and tables 
which would make us realise how mysterious they are; 
we have direct knowledge of time and of the human 
spirit which makes us reject as inadequate that merely 
symbolic conception of the world which is so often mis- 
taken for an insight into its nature. 

The Four-Dimensional World. I do not know whether 
you have been keenly alive to the fact that for some time 
now we have been immersed in a four-dimensional 
world. The fourth dimension required no introduction; 
as soon as we began to consider events it was there. 
Events obviously have a fourfold order which we can 
dissect into right or left, behind or in front, above or 
below, sooner or later — or into many alternative sets of 
fourfold specification. The fourth dimension is not a 
difficult conception. It is not difficult to. conceive of 
events as ordered in four dimensions; it is impossible to 
conceive them otherwise. The trouble begins when we 
continue farther along this line of thought, because by 
long custom we have divided the world of events into 
three-dimensional sections or instants, and regarded the 
piling of the instants as something distinct from a 
dimension. That gives us the usual conception of a 
three-dimensional world floating in the stream of time. 
This pampering of a particular dimension is not entirely 
without foundation; it is our crude appreciation of the 
absolute separation of space-relations and time-relations 
by the hour-glass figures. But the crude discrimination 
has to be replaced by a more accurate discrimination. 



THE FOUR-DIMENSIONAL WORLD 53 

The supposed planes of structure represented bj 
Now lines separated one dimension from the other 
three; but the cones of structure given by the hour- 
glass figures keep the four dimensions firmly pinned 
together.* 

We are accustomed to think of a man apart from his 
duration. When I portrayed "Myself" in Fig. 2, you 
were for the moment surprised that I should include 
my boyhood and old age. But to think of a man without 
his duration is just as abstract as to think of a man 
without his inside. Abstractions are useful, and a man 
without his inside (that is to say, a surface) is a well- 
known geometrical conception. But we ought to realise 
what is an abstraction and what is not. The "four- 
dimensional worms" introduced in this chapter seem to 
many people terribly abstract. Not at all; they are un- 
familiar conceptions but not abstract conceptions. It is 
the section of the worm (the man Now) which is an 
abstraction. And as sections may be taken in somewhat 
different directions, the abstraction is made differently 
by different observers who accordingly attribute different 
FitzGerald contractions to it. The non-abstract man 
enduring through time is the common source from which 
the different abstractions are made. 

The appearance of a four-dimensional world in this 
subject is due to Minkowski. Einstein showed the rela- 
tivity of the familiar quantities of physics; Minkowski 
showed how to recover the absolute by going back to 
their four-dimensional origin and searching more deeply. 

* In Fig. 4 the scale is such that a second of time corresponds to 
70,000 miles of space. If we take a more ordinary scale of experience, 
say a second to a yard, the Seen-Now lines become almost horizontal; 
and it will easily be understood why the cones which pin the four 
dimensions together have generally been mistaken for sections separating 
them. 



54 TIME 

The Velocity of Light. A feature of the relativity 
theory which seems to have aroused special interest 
among philosophers is the absoluteness of the velocity of 
light. In general velocity is relative. If I speak of a 
velocity of 40 kilometres a second I must add "relative 
to the earth", "relative to Arcturus", or whatever refer- 
ence body I have in mind. No one will understand any- 
thing from my statement unless this is added or implied. 
But it is a curious fact that if I speak of a velocity 
of 299,796 kilometres a second it is unnecessary to 
add the explanatory phrase. Relative to what? Rela- 
tive to any and every star or particle of matter in the 
universe. 

It is no use trying to overtake a flash of light; 
however fast you go it is always travelling away from 
you at 186,000 miles a second. Now from one point 
of view this is a rather unworthy deception that Nature 
has practised upon us. Let us take our favourite observer 
who travels at 161,000 miles a second and send him in 
pursuit of the flash of light. It is going 25,000 miles 
a second faster than he is; but that is not what he will 
report. Owing to the contraction of his standard scale 
his miles are only half-miles; owing to the slowing down 
of his clocks his seconds are double-seconds. His 
measurements would therefore make the speed 100,000 
miles a second (really half-miles per double-second). 
He makes a further mistake in synchronising the clocks 
with which he records the velocity. (You will remember 
that he uses a different Now line from ours.). This 
brings the speed up to 186,000 miles a second. From 
his own point of view the traveller is lagging hopelessly 
behind the light; he does not realise what a close race 
he is making of it, because his measuring appliances 
have been upset. You will note that the evasiveness of 



THE VELOCITY OF LIGHT 55 

the light-flash is not in the least analogous to the 
evasiveness of the rainbow. 

But although this explanation may help to reconcile 
us to what at first seems a blank impossibility, it is not 
really the most penetrating. You will remember that 
a Seen-Now line, or track of a flash of light, represents 
the grain of the world-structure. Thus the peculiarity 
of a velocity of 299,796 kilometres a second is that it 
coincides with the grain of the world. The four- 
dimensional worms representing material bodies must 
necessarily run across the grain into the future cone, and 
we have to introduce some kind of reference frame to 
describe their course. But the flash of light is exactly 
along the grain, and there is no need of any artificial 
system of partitions to describe this fact. 

The number 299,796 (kilometres per second) is, 
so to speak, a code-number for the grain of the wood. 
Other code-numbers correspond to the various worm- 
holes which may casually cross the grain. We have 
different codes corresponding to different frames of 
space and time; the code-number of the grain of the 
wood is the only one which is the same in all codes. 
This is no accident; but I do not know that any deep 
inference is to be drawn from it, other than that our 
measure-codes have been planned rationally so as to turn 
on the essential and not on the casual features of world- 
structure. 

The speed of 299,796 kilometres per second which 
occupies a unique position in every measure-system is 
commonly referred to as the speed of light. But it is 
much more than that; it is the speed at which the mass 
of matter becomes infinite, lengths contract to zero, 
clocks stand still. Therefore it crops up in all kinds of 
problems whether light is concerned or not. 



56 TIME 

The scientist's interest in the absoluteness of this 
velocity is very great; the philosopher's interest has 
been, I think, largely a mistaken interest. In asserting 
its absoluteness scientists mean that they have assigned 
the same number to it in every measure-system; but 
that is a private arrangement of their own — an un- 
witting compliment to its universal importance.* Turn- 
ing from the measure-numbers to the thing described 
by them, the "grain" is certainly an absolute feature 
of the wood, but so also are the "worm-holes" 
(material particles). The difference is that the grain is 
essential and universal, the worm-holes casual. Science 
and philosophy have often been at cross-purposes in 
discussing the Absolute — a misunderstanding which is 
I am afraid chiefly the fault of the scientists. In science 
we are chiefly concerned with the absoluteness or relativity 
of the descriptive terms we employ; but when the term 
absolute is used with reference to that which is being 
described it has generally the loose meaning of "uni- 
versal" as opposed to "casual". 

Another point on which there has sometimes been a 
misunderstanding is the existence of a superior limit to 
velocity. It is not permissible to say that no velocity can 
exceed 299,796 kilometres per second. For example, 
imagine a search-light capable of sending an accurately 
parallel beam as far as Neptune. If the search-light is 
made to revolve once a minute, Neptune's end of the 
beam will move round a circle with velocity far greater 
than the above limit. This is an example of our habit 
of creating velocities by a mental association of states 

* In the general relativity theory (chapter vi) measure-systems are 
employed in which the velocity of light is no longer assigned the same 
constant value, but it continues to correspond to the grain of absolute 
world-structure. 



THE VELOCITY OF LIGHT 57 

which are not themselves in direct causal connection. 
The assertion made by the relativity theory is more 
restricted, viz. — 

Neither matter, nor energy, nor anything capable of 
being used as a signal can travel faster than 299,796 
kilometres per second, provided that the velocity is 
referred to one of the frames of space and time con- 
sidered in this chapter.* 

The velocity of light in matter can under certain 
circumstances (in the phenomenon of anomalous dis- 
persion) exceed this value. But the higher velocity is 
only attained after the light has been passing through 
the matter for some moments so as to set the molecules 
in sympathetic vibration. An unheralded light-flash 
travels more slowly. The speed, exceeding 299,796 
kilometres a second, is, so to speak, achieved 
by prearrangement, and has no application in sig- 
nalling. 

We are bound to insist on this limitation of the speed 
of signalling. It has the effect that it is only possible to 
signal into the Absolute Future. The consequences of 
being able to transmit messages concerning events 
Here-Now into the neutral wedge are too bizarre to 
contemplate. Either the part of the neutral wedge that 
can be reached by the signals must be restricted in a 
way which violates the principle of relativity; or it will 
be possible to arrange for a confederate to receive the 
messages which we shall send him to-morrow, and to 
retransmit them to us so that we receive them to-dav^' 
The limit to the velocity of signals is our bulwark 

* Some proviso of this kind is clearly necessary. We often employ 
for special purposes a frame of reference rotating with the earth; in this 
frame the stars describe circles once a day, and are therefore ascribed 
enormous velocities. 



58 TIME 

against that topsy-turvydom of past and future, of which 
Einstein's theory is sometimes wrongfully accused. 

Expressed in the conventional way this limitation of 
the speed of signalling to 299,796 kilometres a 
second seems a rather arbitrary decree of Nature. We 
almost feel it as a challenge to find something that goes 
faster. But if we state it in the absolute form that 
signalling is only possible along a track of temporal 
relation and not along a track of spatial relation the 
restriction seems rational. To violate it we have not 
merely to find something which goes just 1 kilometre 
per second better, but something which overleaps that 
distinction of time and space — which, we are all con- 
vinced, ought to be maintained in any sensible theory. 

Practical Applications. In these lectures I am concerned 
more with the ideas of the new theories than with their 
practical importance for the advancement of science. 
But the drawback of dwelling solely on the underlying 
conceptions is that it is likely to give the impression that 
the new physics is very much u up in the air". That is 
by no means true, and the relativity theory is used in 
a businesslike way in the practical problems to which 
it applies. I can only consider here quite elementary 
problems which scarcely do justice to the power of the 
new theory in advanced scientific research. Two 
examples must suffice. 

1. It has often been suggested that the stars will be 
retarded by the back-pressure of their own radiation. 
The idea is that since the star is moving forward the 
emitted radiation is rather heaped up in front of it and 
thinned out behind. Since radiation exerts pressure the 
pressure will be stronger on the front surface than on 
the rear, Therefore there is a force retarding the star 



PRACTICAL APPLICATIONS 59 

tending to bring it gradually to rest. The effect might 
be of great importance in the study of stellar motions; 
it would mean that on the average old stars must have 
lower speeds than young stars — a conclusion which, as 
it happens, is contrary to observation. 

But according to the theory of relativity "coming to 
rest" has no meaning. A decrease of velocity relative 
to one frame is an increase relative to another frame. 
There is no absolute velocity and no absolute rest for 
the star to come to. The suggestion may therefore be 
at once dismissed as fallacious. 

2. The B particles shot out by radioactive substances 
are electrons travelling at speeds not much below 
the speed of light. Experiment shows that the mass 
of one of these high-speed electrons is considerably 
greater than the mass of an electron at rest. The theory 
of relativity predicts this increase and provides the 
formula for the dependence of mass on velocity. The 
increase arises solely from the fact that mass is a relative 
quantity depending by definition on the relative quan- 
tities length and time. 

Let us look at a 3 particle from its own point of view. 
It is an ordinary electron in no wise different from any 
other. But it is travelling with unusually high speed? 
"No", says the electron, "That is your point of view. 
I contemplate with amazement your extraordinary 
speed of 100,000 miles a second with which you are 
shooting past me. I wonder what it feels like to move 
so quickly. However, it is no business of mine." So 
the p particle, smugly thinking itself at rest, pays no 
attention to our goings on, and arranges itself with the 
usual mass, radius and charge. It has just the standard 
mass of an electron, 9.10" 28 grams. But mass and 
radius are relative quantities, and in this case the frame 



60 TIME 

to which they are referred is evidently the frame appro- 
priate to an electron engaged in self-contemplation, viz. 
the frame in which it is at rest But when we talk about 
mass we refer it to the frame in which we are at rest. 
By the geometry of the four-dimensional world we can 
calculate the formulae for the change of reckoning of 
mass in two different frames, which is consequential on 
the change of reckoning of length and time; we find 
in fact that the mass is increased in the same ratio as the 
length is diminished (FitzGerald factor). The increase 
of mass that we observe arises from the change of 
reckoning between the electron's own frame and our 
frame. 

All electrons are alike from their own point of view. 
The apparent differences arise in fitting them into our 
own frame of reference which is irrelevant to their 
structure. Our reckoning of their mass is higher than 
their own reckoning, and increases with the difference 
between our respective frames, i.e. with the relative 
velocity between us. 

We do not bring forward these results to demonstrate 
or confirm the truth of the theory, but to show the use 
of the theory. They can both be deduced from the 
classical electromagnetic theory of Maxwell coupled (in 
the second problem) with certain plausible assumptions 
as to the conditions holding at the surface of an electron. 
But to realise the advantage of the new theory we must 
consider not what could have been but what was deduced 
from the classical theory. The historical fact is that the 
conclusions of the classical theory as to the first prob- 
lem were wrong; an important compensating factor 
escaped notice. Its conclusions as to the second problem 
were (after some false starts) entirely correct numer- 
ically. But since the result was deduced from the electro- 



SUMMARY 61 

magnetic equations of the electron it was thought that 
it depended on the fact that an electron is an electrical 
structure; and the agreement with observation was 
believed to confirm the hypothesis that an electron is 
pure electricity and nothing else. Our treatment above 
makes no reference to any electrical properties of the 
electron, the phenomenon having been found to arise 
solely from the relativity of mass. Hence, although 
there may be other good reasons for believing that an 
electron consists solely of negative electricity, the in- 
crease of mass with velocity is no evidence one way or 
the other. 

In this chapter the idea of a multiplicity of frames of 
space has been extended to a multiplicity of frames of 
space and time. The system of location in space, called 
a frame of space, is only a part of a fuller system of 
location of events in space and time. Nature provides 
no indication that one of these frames is to be preferred 
to the others. The particular frame in which we are 
relatively at rest has a symmetry with respect to us 
which other frames do not possess, and for this reason 
we have drifted into the common assumption that it is 
the only reasonable and proper frame; but this egocen- 
tric outlook should now be abandoned, and all frames 
treated as on the same footing. By considering time 
and space together we have been able to understand 
how the multiplicity of frames arises. They correspond 
to different directions of section of the four-dimensional 
world of events, the sections being the "world-wide 
instants". Simultaneity (Now) is seen to be relative. 
The denial of absolute simultaneity is intimately con- 
nected with the denial of absolute velocity; knowledge 
of absolute velocity would enable us to assert that 



62 TIME 

certain events in the past or future occur Here but not 
Now; knowledge of absolute simultaneity would tell us 
that certain events occur Now but not Here. Removing 
these artificial sections, we have had a glimpse of the 
absolute world-structure with its grain diverging and 
interlacing after the plan of the hour-glass figures. By 
reference to this structure we discern an absolute dis- 
tinction between space-like and time-like separation of 
events — a distinction which justifies and explains our 
instinctive feeling that space and time are fundamentally 
different. Many of the important applications of the 
new conceptions to the practical problems of physics 
are too technical to be considered in this book; one of 
the simpler applications is to determine the changes of 
the physical properties of objects due to rapid motion. 
Since the motion can equally well be described as a 
motion of ourselves relative to the object or of the 
object relative to ourselves, it cannot influence the abso- 
lute behaviour of the object. The apparent changes in 
the length, mass, electric and magnetic fields, period of 
vibration, etc., are merely a change of reckoning intro- 
duced in passing from the frame in which the object is 
at rest to the frame in which the observer is at rest. 
Formulae for calculating the change of reckoning of 
any of these quantities are easily deduced now that the 
geometrical relation of the frames has been ascertained. 



Chapter IV 

THE RUNNING-DOWN OF THE UNIVERSE 

Shuffling. The modern outlook on the physical world is 
not composed exclusively of conceptions which have 
arisen in the last twenty-five years; and we have now 
to deal with a group of ideas dating far back in the last 
century which have not essentially altered since the 
time of Boltzmann. These ideas display great activity 
and development at the present time. The subject is 
relevant at this stage because it has a bearing on the 
deeper aspects of the problem of Time; but it is so 
fundamental in physical theory that we should be bound 
to deal with it sooner or later in any comprehensive 
survey. 

If you take a pack of cards as it comes from the 
maker and shuffle it for a few minutes, all trace of 
the original systematic order disappears. The order wiil 
never come back however long you shuffle. Something 
has been done which cannot be undone, namely, the intro- 
duction of a random element in place of arrangement. 

Illustrations may be useful even when imperfect, and 
therefore I have slurred over two points, which affect 
the illustration rather than the application which we are 
about to make. It was scarcely true to say that the 
shuffling cannot be undone. You can sort out the cards 
into their original order if you like. But in considering 
the shuffling which occurs in the physical world we are 
not troubled by a deus ex machina like you. I am not 
prepared to say how far the human mind is bound by the 
conclusions we shall reach. So I exclude you — at least 
I exclude that activity of your mind which you employ 

63 



64 THE RUNNING-DOWN OF THE UNIVERSE 

in sorting the cards. I allow you to shuffle them because 
you can do that absent-mindedly. 

Secondly, it is not quite true that the original order 
never comes back. There is a ghost of a chance that 
some day a thoroughly shuffled pack will be found to 
have come back to the original order. That is because 
of the comparatively small number of cards in the pack. 
In our applications the units are so numerous that this 
kind of contingency can be disregarded. 

We shall put forward the contention that — 

Whenever anything happens which cannot be undone, 
it is always reducible to the introduction of a random 
element analogous to that introduced by shuffling. 

Shuffling is the only thing which Nature cannot undo. 
When Humpty Dumpty had a great fall — 

All the king's horses and all the king's men 
Cannot put Humpty Dumpty together again. 

Something had happened which could not be undone. 
The fall could have been undone. It is not necessary to 
invoke the king's horses and the king's men; if there 
had been a perfectly elastic mat underneath, that would 
have sufficed. At the end of his fall Humpty Dumpty 
had kinetic energy which, properly directed, was just 
sufficient to bounce him back on to the wall again. But, 
the elastic mat being absent, an irrevocable event hap- 
pened at the end of the fall — namely, the introduction 
of a random element into Humpty Dumpty. 

But why should we suppose that shuffling is the only 
process that cannot be undone? 

The Moving Finger writes; and, having writ, 
Moves on: nor all thy Piety and Wit 
Can lure it back to cancel half a Line. 



SHUFFLING 65 

When there is no shuffling, is the Moving Finger stayed? 
The answer of physics is unhesitatingly Yes. To judge 
of this we must examine those operations of Nature in 
which no increase of the random element can possibly 
occur. These fall into two groups. Firstly, we can 
study those laws of Nature which control the behaviour 
of a single unit. Clearly no shuffling can occur in these 
problems; you cannot take the King of Spades away 
from the pack and shuffle him. Secondly, we can study 
the processes of Nature in a crowd which is already so 
completely shuffled that there is no room for any further 
increase of the random element. If our contention is 
right, everything that occurs in these conditions is 
capable of being undone. We shall consider the first 
condition immediately; the second must be deferred 
until p. 78. 

Any change occurring to a body which can be treated 
as a single unit can be undone. The laws of Nature 
admit of the undoing as easily as of the doing. The 
earth describing its orbit is controlled by laws of motion 
and of gravitation; these admit of the earth's actual mo- 
tion, but they also admit of the precisely opposite 
motion. In the same field of force the earth could retrace 
its steps; it merely depends on how it was started off. It 
may be objected that we have no right to dismiss the 
starting-off as an inessential part of the problem; it may 
be as much a part of the coherent scheme of Nature as 
the laws controlling the subsequent motion. Indeed, as- 
tronomers have theories explaining why the eight planets 
all started to move the same way round the sun. But 
that is a problem of eight planets, not of a single 
individual — a problem of the pack, not of the isolated 
card. So long as the earth's motion is treated as an 
isolated problem, no one would dream of putting into 



66 THE RUNNING-DOWN OF THE UNIVERSE 

the laws of Nature a clause requiring that it must go 
this way round and not the opposite. 

There is a similar reversibility of motion in fields of 
electric and magnetic force. Another illustration can be 
given from atomic physics. The quantum laws admit 
of the emission of certain kinds and quantities of light 
from an atom; these laws also admit of absorption of 
the same kinds and quantities, i.e. the undoing of the 
emission. I apologise for an apparent poverty of illus- 
tration; it must be remembered that many properties of 
a body, e.g. temperature, refer to its constitution as a 
large number of separate atoms, and therefore the laws 
controlling temperature cannot be regarded as control- 
ling the behaviour of a single individual. 

The common property possessed by laws governing 
the individual can be stated more clearly by a reference 
to time. A certain sequence of states running from past 
to future is the doing of an event; the same sequence 
running from future to past is the undoing of it — be- 
cause in the latter case we turn round the sequence so as 
to view it in the accustomed manner from past to future. 
So if the laws of Nature are indifferent as to the doing 
and undoing of an event, they must be indifferent as 
to a direction of time from past to future. That is their 
common feature, and it is seen at once when (as 
usual) the laws are formulated mathematically. There 
is no more distinction between past and future than 
between right and left. In algebraic symbolism, left is 
— x, right is +*; past is — f, future is +J. This holds 
for all laws of Nature governing the behaviour of non- 
composite individuals — the "primary laws", as we shall 
call them. There is only one law of Nature — the second 
law of thermodynamics — which recognises a distinction 
between past and future more profound than the 



SHUFFLING 67 

difference of plus and minus. It stands aloof from all 
the rest. But this law has no application to the behaviour 
of a single individual, and as we shall see later its sub- 
ject-matter is the random element in a crowd. 

Whatever the primary laws of physics may say, it is 
obvious to ordinary experience that there is a distinction 
between past and future of a different kind from the 
distinction of left and right. In The Plattner Story 
H. G. Wells relates how a man strayed into the fourth 
dimension and returned with left and right interchanged. 
But we notice that this interchange is not the theme of 
the story; it is merely a corroborative detail to give an 
air of verisimilitude to the adventure. In itself the 
change is so trivial that even Mr. Wells cannot weave 
a romance out of it. But if the man had come back with 
past and future interchanged, then indeed the situation 
would have been lively. Mr. Wells in The Time-Machine 
and Lewis Carroll in Sylvie and Bruno give us a glimpse 
of the absurdities which occur when time runs back- 
wards. If space is "looking-glassed" the world con- 
tinues to make sense; but looking-glassed time has an 
inherent absurdity which turns the world-drama into 
the most nonsensical farce. 

Now the primary laws of physics taken one by one 
all declare that they are entirely indifferent as to which 
way you consider time to be progressing, just as they 
are indifferent as to whether you view the world from 
the right or the left. This -is true of the classical laws, 
the relativity laws, and even of the quantum laws. It 
is not an accidental property; the reversibility is inherent 
in the whole conceptual scheme in which these laws 
find a place. Thus the question whether the world does 
or does not "make sense" is outside the range of these 
laws. We have to appeal to the one outstanding law — 



68 THE RUNNING-DOWN OF THE UNIVERSE 

the second law of thermodynamics — to put some sense 
into the world. It opens up a new province of know- 
ledge, namely, the study of organisation; and it is in con- 
nection with organisation that a direction of time-flow 
and a distinction between doing and undoing appears for 
the first time. 



Time's Arrow, The great thing about time is that it goes 
on. But this is an aspect of it which the physicist some- 
times seems inclined to neglect. In the four-dimensional 
world considered in the last chapter the events past and 
future lie spread out before us as in a map. The events 
are there in their proper spatial and temporal relation; 
but there is no indication that they undergo what has 
been described as "the formality of taking place", and 
the question of their doing or undoing does not arise. 
We see in the map the path from past to future or from 
future to past; but there is no signboard to indicate that 
it is a one-way street. Something must be added to 
the geometrical conceptions comprised in Minkowski's 
world before it becomes a complete picture of the world 
as we know it. We may appeal to consciousness to suffuse 
the whole — to turn existence into happening, being into 
becoming. But first let us note that the picture as it 
stands is entirely adequate to represent those primary 
laws of Nature which, as we have seen, are indifferent 
to a direction of time. Objection has sometimes been 
felt to the relativity theory because its four-dimensional 
picture of the world seems to overlook the directed 
character of time. The objection is scarcely logical, for 
the theory is in this respect no better and no worse than 
its predecessors. The classical physicist has been using 
without misgiving a system of laws which do not 



TIME'S ARROW 69 

recognise a directed time; he is shocked that the new 
picture should expose this so glaringly. 

Without any mystic appeal to consciousness it is 
possible to find a direction of time on the four-dimen- 
sional map by a study of organisation. Let us draw an 
arrow arbitrarily. If as we follow the arrow we find 
more and more of the random element in the state of 
the world, then the arrow is pointing towards the future; 
if the random element decreases the arrow points 
towards the past. That is the only distinction known to 
physics. This follows at once if our fundamental con- 
tention is admitted that the introduction of randomness 
is the only thing which cannot be undone. 

I shall use the phrase "time's arrow" to express this 
one-way property of time which has no analogue in 
space. It is a singularly interesting property from a 
philosophical standpoint. We must note that — 

( 1 ) It is vividly recognised by consciousness. 

(2) It is equally insisted on by our reasoning faculty, 
which tells us that a reversal of the arrow would render 
the external world nonsensical. 

(3) It makes no appearance in physical science except 
in the study of organisation of a number of individuals. 
Here the arrow indicates the direction of progressive 
increase of the random element. 

Let us now consider in detail how a random element 
brings the irrevocable into the world. When a stone 
falls it acquires kinetic energy, and the amount of the 
energy is just that which would be required to lift the 
stone back to its original height. By suitable arrange- 
ments the kinetic energy can be made to perform this 
task; for example, if the stone is tied to a string it can 
alternately fall and reascend like a pendulum. But if 
the stone hits an obstacle its kinetic energy is converted 



70 THE RUNNING-DOWN OF THE UNIVERSE 

into heat-energy. There is still the same quantity of 
energy, but even if we could scrape it together and put 
it through an engine we could not lift the stone back 
with it. What has happened to make the energy no 
longer serviceable? 

Looking microscopically at the falling stone we see 
an enormous multitude of molecules moving downwards 
with equal and parallel velocities — an organised motion 
like the march of a regiment. We have to notice two 
things, the energy and the organisation of the energy. 
To return to its original height the stone must preserve 
both of them. 

When the stone falls on a sufficiently elastic surface 
the motion may be reversed without destroying the 
organisation. Each molecule is turned backwards and the 
whole array retires in good order to the starting-point — 

The famous Duke of York 

With twenty thousand men, 
He marched them up to the top of the hill 

And marched them down again. 

History is not made that way. But what usually happens 
at the impact is that the molecules suffer more or less 
random collisions and rebound in all directions. They 
no longer conspire to make progress in any one direc- 
tion; they have lost their organisation. Afterwards they 
continue to collide with one another and keep changing 
their directions of motion, but they never again find a 
common purpose. Organisation cannot be brought 
about by continued shuffling. And so, although the 
energy remains quantitatively sufficient (apart from un- 
avoidable leakage which we suppose made good), it 
cannot lift the stone back. To restore the stone we must 
supply extraneous energy which has the required 
amount of organisation. 



COINCIDENCES 71 

Here a point arises which unfortunately has no 
analogy in the shuffling of a pack of cards. No one 
(except a conjurer) can throw two half-shuffled packs 
into a hat and draw out one pack in its original order 
and one pack fully shuffled. But we can and do put 
partly disorganised energy into a steam-engine, and 
draw it out again partly as fully organised energy of 
motion of massive bodies and partly as heat-energy in 
a state of still worse disorganisation. Organisation of 
energy is negotiable, and so is the disorganisation or 
random element; disorganisation does not for ever 
remain attached to the particular store of energy which 
first suffered it, but may be passed on elsewhere. We 
cannot here enter into the question why there should be 
a difference between the shuffling of energy and the 
shuffling of material objects; but it is necessary to use 
some caution in applying the analogy on account of this 
difference. As regards heat-energy the temperature is 
the measure of its degree of organisation; the lower the 
temperature, the greater the disorganisation. 

Coincidences, There are such things as chance coinci- 
dences; that is to say, chance can deceive us by bringing 
about conditions which look very unlike chance. In 
particular chance might imitate organisation, whereas 
we have taken organisation to be the antithesis o{ 
chance or, as we have called it, the "random element". 
This threat to our conclusions is, however, not very 
serious. There is safety in numbers. 

Suppose that you have a vessel divided by a partition 
into two halves, one compartment containing air and 
the other empty. You withdraw the partition. For the 
moment all the molecules of air are in one half of the 
vessel; a fraction of a second later they are spread over 



72 THE RUNNING-DOWN OF THE UNIVERSE 

the whole vessel and remain so ever afterwards. The 
molecules will not return to one half of the vessel; the 
spreading cannot be undone — unless other material is 
introduced into the problem to serve as a scapegoat for 
the disorganisation and carry off the random element 
elsewhere. This occurrence can serve as a criterion to 
distinguish past and future time. If you observe first 
the molecules spread through the vessel and (as it 
seems to you) an instant later the molecules all in one 
half of it — then your consciousness is going backwards, 
and you had better consult a doctor. 

Now each molecule is wandering round the vessel 
with no preference for one part rather than the other. 
On the average it spends half its time in one compart- 
ment and half in the other. There is a faint possibility 
that at one moment all the molecules might in this way 
happen to be visiting the one half of the vessel. You 
will easily calculate that if n is the number of molecules 
(roughly a quadrillion) the chance of this happening is 
( y 2 ) n . The reason why we ignore this chance may be seen 
by a rather classical illustration. If I let my fingers 
wander idly over the keys of a typewriter it might 
happen that my screed made an intelligible sentence. 
If an army of monkeys were strumming on typewriters 
they might write all the books in the British Museum. 
The chance of their doing so is decidedly more favour- 
able than the chance of the molecules returning to one 
half of the vessel. 

When numbers are large, chance is the best warrant 
for certainty. Happily in the study of molecules and 
energy and radiation in bulk we have to deal with a 
vast population, and we reach a certainty which does 
not always reward the expectations of those who court 
the fickle goddess. 



COINCIDENCES 73 

In one sense the chance of the molecules returning 
to one half of the vessel is too absurdly small to think 
about. Yet in science we think about it a great deal, 
because it gives a measure of the irrevocable mischief 
we did when we casually removed the partition. Even 
if we had good reasons for wanting the gas to fill the 
vessel there was no need to waste the organisation; as 
we have mentioned, it is negotiable and might have been 
passed on somewhere where it was useful.* When the 
gas was released and began to spread across the vessel, 
say from left to right, there was no immediate increase 
of the random element. In order to spread from left to 
right, left-to-right velocities of the molecules must have 
preponderated, that is to say the motion was partly 
organised. Organisation of position was replaced by 
organisation of motion. A moment later the molecules 
struck the farther wall of the vessel and the random 
element began to increase. But, before it was destroyed, 
the left-to-right organisation of molecular velocities was 
the exact numerical equivalent of the lost organisation 
in space. By that we mean that the chance against the 
left-to-right preponderance of velocity occurring by 
accident is the same as the chance against segregation 
in one half of the vessel occurring by accident. 

The adverse chance here mentioned is a preposterous 
number which (written in the usual decimal notation) 
would fill all the books in the world many times over. 
We are not interested in it as a practical contingency; 
but we are interested in the fact that it is definite. It 
raises "organisation" from a vague descriptive epithet 
to one of the measurable quantities of exact science. 
We are confronted with many kinds of organisation. 

* If the gas in expanding had been made to move a piston, the 
organisation would have passed into the motion of the piston. 



74 THE RUNNING-DOWN OF THE UNIVERSE 

The uniform march of a regiment is not the only form 
of organised motion; the organised evolutions of a 
stage chorus have their natural analogue in sound waves. 
A common measure can now be applied to all forms 
of organisation. Any loss of organisation is equitably 
measured by the chance against its recovery by an acci- 
dental coincidence. The chance is absurd regarded as 
a contingency, but it is precise as a measure. 

The practical measure of the random element which 
can increase in the universe but can never decrease is 
called entropy. Measuring by entropy is the same as 
measuring by the chance explained in the last paragraph, 
only the unmanageably large numbers are transformed 
(by a simple formula) into a more convenient scale of 
reckoning. Entropy continually increases. We can, 
by isolating parts of the world and postulating rather 
idealised conditions in our problems, arrest the increase, 
but we cannot turn it into a decrease. That would 
involve something much worse than a violation of an 
ordinary law of Nature, namely, an improbable coinci- 
dence. The law that entropy always increases — the 
second law of thermodynamics — holds, I think, the 
supreme position among the laws of Nature. If someone 
points out to you that your pet theory of the universe 
is in disagreement with Maxwell's equations — then so 
much the worse for Maxwell's equations. If it is found 
to be contradicted by observation — well, these experi- 
mentalists do bungle things sometimes. But if your 
theory is found to be against the second law of thermo- 
dynamics I can give you no hope; there is nothing for 
it but to collapse in deepest humiliation. This exaltation 
of the second law is not unreasonable. There are other 
laws which we have strong reason to believe in, and we 
feel that a hypothesis which violates them is highly 



PRIMARY AND SECONDARY LAW] 75 

improbable; but the improbability is vague and does 
not confront us as a paralysing array of figures, whereas 
the chance against a breach of the second law (i.e. 
against a decrease of the random element) can be stated 
in figures which are overwhelming. 

I wish I could convey to you the amazing power of 
this conception of entropy in scientific research. From 
the property that entropy must always increase, practical 
methods of measuring it have been found. The chain 
of deductions from this simple law have been almost 
illimitable; and it has been equally successful in con- 
nection with the most recondite problems of theoretical 
physics and the practical tasks of the engineer. Its 
special feature is that the conclusions are independent 
of the nature of the microscopical processes that are 
going on. It is not concerned with the nature of the 
individual; it is interested in him only as a component 
of a crowd. Therefore the method is applicable in 
fields of research where our ignorance has scarcely begun 
to lift, and we have no hesitation in applying it to prob- 
lems of the quantum theory, although the mechanism 
of the individual quantum process is unknown and at 
present unimaginable. 

Primary and Secondary Law. I have called the laws 
controlling the behaviour of single individuals "primary 
laws", implying that the second law of thermodynamics, 
although a recognised law of Nature, is in some sense a 
secondary law. This distinction can now be placed on 
a regular footing. Some things never happen in the 
physical world because they are impossible; others 
because they are too improbable. The laws which forbid 
the first are the primary laws; the laws which forbid the 
second are the secondary laws. It has been the convic- 



76 THE RUNNING-DOWN OF THE UNIVERSE 

tion of nearly all physicists* that at the root of every- 
thing there is a complete scheme of primary law govern- 
ing the career of every particle or constituent of the 
world with an iron determinism. This primary scheme 
is all-sufficing, for, since it fixes the history of every 
constituent of the world, it fixes the whole world- 
history. 

But for all its completeness primary law does not 
answer every question about Nature which we might 
reasonably wish to put. Can a universe evolve back- 
wards, i.e. develop in the opposite way to our own 
system? Primary law, being indifferent to a time- 
direction, replies, "Yes, it is not impossible". Secondary 
law replies, "No, it is too improbable". The answers are 
not really in conflict; but the first, though true, rather 
misses the point. This is typical of some much more 
commonplace queries. If I put this saucepan of water 
on this fire, will the water boil? Primary law can answer 
definitely if it is given the chance; but it must be 
understood that "this" translated into mathematics 
means a specification of the positions, motions, etc., of 
some quadrillions of particles and elements of energy. 
So in practice the question answered is not quite the 
one that; is asked: If I put a saucepan resembling this 
one in a few major respects on a fire, will the water 
boil? Primary law replies, "It may boil; it may freeze; it 
may do pretty well anything. The details given are insuf- 
ficient to exclude any result as impossible." Secondary 
law replies plainly, "It will boil because it is too im- 
probable that it should do anything else." Secondary 
law is not in conflict with primary law, nor can we re- 
gard it as essential to complete a scheme of law already 

♦There are, however, others beside myself who have recently begun to 
question it. 



THERMODYNAMICAL EQUILIBRIUM 77 

complete in itself. It results from a different '(and 
rather more practical) conception of the aim of our 
traffic with the secrets of Nature. 

The question whether the second law of thermo- 
dynamics and other statistical laws are mathematical 
deductions from the primary laws, presenting their 
results in a conveniently usable form, is difficult to 
answer; but I think it is generally considered that there 
is an unbridgeable hiatus. At the bottom of all the 
questions settled by secondary law there is an elusive 
conception of u a priori probability of states of the 
world" which involves an essentially different attitude to 
knowledge from that presupposed in the construction of 
the scheme of primary law. 

Thermo dynamical Equilibrium. Progress of time intro- 
duces more and more of the random element into the 
constitution of the world. There is less of chance about 
the physical universe to-day than there will be to-mor- 
row. It is curious that in this very matter-of-fact 
branch of physics, developed primarily because of its 
importance for engineers, we can scarcely avoid express- 
ing ourselves in teleological language. We admit that 
the world contains both chance and design, or at any 
rate chance and the antithesis of chance. This antithe- 
sis is emphasised by our method of measurement of 
entropy; we assign to the organisation or non-chance 
element a measure which is,* so to speak, proportional 
to the strength of our disbelief in a chance origin for it. 
"A fortuitous concourse of atoms" — that bugbear of 
the theologian — has a very harmless place in orthodox 
physics. The physicist is acquainted with it as a much- 
prized rarity. Its properties are very distinctive, and 
unlike those of the physical world in general. The 



78 THE RUNNING-DOWN OF THE UNIVERSE 

scientific name for a fortuitous concourse of atoms is 
"thermodynamical equilibrium". 

Thermodynamical equilibrium is the other case 
which we promised to consider in which no increase in 
the random element can occur, namely, that in which the 
shuffling is already as thorough as possible. We must 
isolate a region of the universe, arranging that no 
energy can enter or leave it, or at least that any bound- 
ary effects are precisely compensated. The conditions 
are ideal, but they can be reproduced with sufficient 
approximation to make the ideal problem relevant to 
practical experiment. A region in the deep interior of 
a star is an almost perfect example of thermodynamical 
equilibrium. Under these isolated conditions the energy 
will be shuffled as it is bandied from matter to aether 
and back again, and very soon the shuffling will be 
complete. 

The possibility of the shuffling becoming complete 
is significant. If after shuffling the pack you tear each 
card in two, a further shuffling of the half-cards becomes 
possible. Tear the cards again and again; each time 
there is further scope for the random element to 
increase. With infinite divisibility there can be no end 
to the shuffling. The experimental fact that a definite 
state of equilibrium is rapidly reached indicates that 
energy is not infinitely divisible, or at least that it is not 
infinitely divided in the natural processes of shuffling. 
Historically this is the result from which the quantum 
theory first arose. We shall return to it in a later 
chapter. 

In such a region we lose time's arrow. You remember 
that the arrow points in the direction of increase of 
the random element. When the random element has 
reached its limit and become steady the arrow does not 



THERMODYNAMICAL EQUILIBRIUM 79 

know which way to point. It would not be true to say 
that such a region is timeless; the atoms vibrate as usual 
like little clocks; by them we can measure speeds and 
durations. Time is still there and retains its ordinary 
properties, but it has lost its arrow; like space it extends, 
but it does not "go on". 

This raises the important question, Is the random 
element (measured by the criterion of probability 
already discussed) the only feature of the physical world 
which can furnish time with an arrow? Up to the 
present we have concluded that no arrow can be found 
from the behaviour of isolated individuals, but there is 
scope for further search among the properties of crowds 
beyond the property represented by entropy. To give 
an illustration which is perhaps not quite so fantastic 
as it sounds, Might not the assemblage become more 
and more beautiful (according to some agreed aesthetic 
standard) as time proceeds?* The question is answered 
by another important law of Nature which runs — 

Nothing in the statistics of an assemblage can distin- 
guish a direction of time when entropy fails to distinguish 
one. 

I think that although this law was only discovered in 
the last few years there is no serious doubt as to its 
truth. It is accepted as fundamental in all modern 
studies of atoms and radiation and has proved to be one 
of the most powerful weapons of progress in such 
researches. It is, of course, one of the secondary laws. 
It does not seem to be rigorously deducible from the 
second law of thermodynamics, and presumably must 
be regarded as an additional secondary law.t 

* In a kaleidoscope the shuffling is soon complete and all the patterns 
are equal as regards random element, but they differ greatly in elegance. 

f The law is so much disguised in the above enunciation that I must 
explain to the advanced reader that I am referring to "the Principle of 



80 THE RUNNING-DOWN OF THE UNIVERSE 

The conclusion is that whereas other statistical 
characters besides entropy might perhaps be used to 
discriminate time's arrow, they can only succeed when 
it succeeds and they fail when it fails. Therefore they 
cannot be regarded as independent tests. So far as 
physics is concerned time's arrow is a property of 
entropy alone. 

Are Space and Time Infinite? I suppose that everyone 
has at some time plagued his imagination with the 
question, Is there an end to space? If space comes to 
an end, what is beyond the end? On the other hand the 
idea that there is no end, but space beyond space for 
ever, is inconceivable. And so the imagination is tossed 
to and fro in a dilemma. Prior to the relativity theory 
the orthodox view was that space is infinite. No one 
can conceive infinite space; w r e had to be content to 
admit in the physical world an inconceivable concep- 
tion — disquieting but not necessarily illogical. Einstein's 
theory now offers a way out of the dilemma. Is space 
infinite, or does it come to an end? Neither. Space 
is finite but it has no end; "finite but unbounded" 
is the usual phrase. 

Infinite space cannot be conceived by anybody; 
finite but unbounded space is difficult to conceive but 
not impossible. I shall not expect you to conceive it; 
but you can try. Think first of a circle; or, rather, not 



Detailed Balancing." This principle asserts that to every type of process 
(however minutely particularised) there is a converse process, and in 
thermodynamical equilibrium direct and converse processes occur with 
equal frequency. Thus every statistical enumeration of the processes is 
unaltered by reversing the time-direction, i.e. interchanging direct and 
converse processes. Hence there can be no statistical criterion for a 
direction of time when there is thermodynamical equilibrium, i.e. when 
entropy is steady and ceases to indicate time's arrow. 



ARE SPACE AND TIME INFINITE? 81 

the circle, but the line forming its circumference. This 
is a finite but endless line. Next think of a sphere — the 
surface of a sphere — that also is a region which is 
finite but unbounded. The surface of this earth never 
comes to a boundary; there is always some country 
beyond the point you have reached; all the same there 
is not an infinite amount of room on the earth. Now go 
one dimension more; circle, sphere — the next thing. 
Got that? Now for the real difficulty. Keep a tight hold 
of the skin of this hypersphere and imagine that the 
inside is not there at all — that the skin exists without 
the inside. That is finite but unbounded space. 

No; I don't think you have quite kept hold of the 
conception. You overbalanced just at the end. It was 
not the adding of one more dimension that was the real 
difficulty; it was the final taking away of a dimension 
that did it. I will tell you what is stopping you. You 
are using a conception of space which must have 
originated many million years ago and has become 
rather firmly embedded in human thought. But the 
space of physics ought not to be dominated by this 
creation of the dawning mind of an enterprising ape. 
Space is not necessarily like this conception; it is like — 
whatever we find from experiment it is like. Now the 
features of space which we discover by experiment are 
extensions, i.e. lengths and distances. So space is like 
a network of distances. Distances are linkages whose 
intrinsic nature is inscrutable; we do not deny the 
inscrutability when we apply measure numbers to them 
— 2 yards, 5 miles, etc. — as a kind of code distinction. 
We cannot predict out of our inner consciousness the 
laws by which code-numbers are distributed among the 
different linkages of the network, any more than we 
can predict how the code-numbers for electromagnetic 



82 THE RUNNING-DOWN OF THE UNIVERSE 

force are distributed. Both are a matter for experi- 
ment. 

If we go a very long way to a point A in one direction 
through the universe and a very long way to a point B 
in the opposite direction, it is believed that between 
A and B there exists a linkage of the kind indicated by 
a very small code-number; in other words these points 
reached by travelling vast distances in opposite directions 
would be found experimentally to be close together. 
Why not? This happens when we travel east and west 
on the earth. It is true that our traditional inflexible 
conception of space refuses to admit it; but there was 
once a traditional conception of the earth which refused 
to admit circumnavigation. In our approach to the 
conception of spherical space the difficult part was to 
destroy the inside of the hypersphere leaving only its 
three-dimensional surface existing. I do not think that 
is so difficult when we conceive space as a network of 
distances. The network over the surface constitutes a 
self-supporting system of linkage which can be con- 
templated without reference to extraneous linkages. We 
can knock away the constructional scaffolding which 
helped us to approach the conception of this kind of 
network of distances without endangering the con- 
ception. 

We must realise that a scheme of distribution of 
inscrutable relations linking points to one another is not 
bound to follow any particular preconceived plan, so 
that there can be no obstacle to the acceptance of any 
scheme indicated by experiment. 

We do not yet know what is the radius of spherical 
space; it must, of course, be exceedingly great com- 
pared with ordinary standards. On rather insecure 
evidence it has been estimated to be not many times 



ARE SPACE AND TIME INFINITE? 83 

greater than the distance of the furthest known nebulae. 
But the boundlessness has nothing to do with the 
bigness. Space is boundless by re-entrant form not by 
great extension. That which is is a shell floating in the 
infinitude of that which is not. We say with Hamlet, 
"I could be bounded in a nutshell and count myself a 
king of infinite space". 

But the nightmare of infinity still arises in regard to 
time. The world is closed in its space dimensions like 
a sphere, but it is open at both ends in the time dimen- 
sion. There is a bending round by which East ulti- 
mately becomes West, but no bending by which Before 
ultimately becomes After. 

I am not sure that I am logical but I cannot feel the 
difficulty of an infinite future time very seriously. The 
difficulty about A.D. co will not happen until we reach 
A.D. 00, and presumably in order to reach A.D. 00 the 
difficulty must first have been surmounted. It should 
also be noted that according to the second law of thermo- 
dynamics the whole universe will reach thermodynamical 
equilibrium at a not infinitely remote date in the future. 
Time's arrow will then be lost altogether and the whole 
conception of progress towards a future fades away. 

But the difficulty of an infinite past is appalling. It 
is inconceivable that we are the heirs of an infinite time 
of preparation; it is not less inconceivable that there was 
once a moment with no moment preceding it. 

This dilemma of the beginning of time would worry 
us more were it not shut out by another overwhelming 
difficulty lying between us and the infinite past. We 
have been studying the running-down of the universe; 
if our views are right, somewhere between the beginning 
of time and the present day we must place the winding 
up of the universe. 



$4 THE RUNNING-DOWN OF THE UNIVERSE 

Travelling backwards into the past we find a world 
with more and more organisation. If there is no barrier 
to stop us earlier we must reach a moment when the 
energy of the world was wholly organised with none of 
the random element in it. It is impossible to go back 
any further under the present system of natural law. 
I do not think the phrase "wholly organised" begs the 
question. The organisation, we are concerned with is 
exactly definable, and there is a limit at which it becomes 
perfect. There is not an infinite series of states of 
higher and still higher organisation; nor, I think, is the 
limit one which is ultimately approached more and more 
slowly. Complete organisation does not tend to be 
more immune from loss than incomplete organisation. 

There is no doubt that the scheme of physics as it 
has stood for the last three-quarters of a century postu- 
lates a date at which either the entities of the universe 
were created in a state of high organisation, or pre- 
existing entities were endowed with that organisation 
which they have been squandering ever since. More- 
over, this organisation is admittedly the antithesis of 
chance. It is something which could not occur for- 
tuitously. 

This has long been used as an argument against a 
too aggressive materialism. It has been quoted as 
scientific proof of the intervention of the Creator at a 
time not infinitely remote from to-day. But I am not 
advocating that we drew any hasty conclusions from it. 
Scientists and theologians alike must regard as some- 
what crude the naive theological doctrine which 
(suitably disguised) is at present to be found in every 
textbook of thermodynamics, namely that some billions 
of years ago God wound up the material universe and 
has left it to chance ever since. This should be regarded 



ARE SPACE AND TIME INFINITE? 85 

as the working-hypothesis of thermodynamics rather 
than its declaration of faith. It is one of those conclu- 
sions from which we can see no logical escape — only 
it suffers from the drawback that it is incredible. As a 
scientist I simply do not believe that the present order 
of things started off with a bang; unscientifically I feel 
equally unwilling to accept the implied discontinuity in 
the divine nature. But I can make no suggestion to 
evade the deadlock. 

Turning again to the other end of time, there is one 
school of thought which finds very repugnant the idea 
of a wearing out of the world. This school is attracted 
by various theories of rejuvenescence. Its mascot is the 
Phoenix. Stars grow cold and die out. May not two 
dead stars collide, and be turned by the energy of the 
shock into fiery vapour from which a new sun — with 
planets and with life — is born? This theory very 
prevalent in the last century is no longer contemplated 
seriously by astronomers. There is evidence that the 
present stars at any rate are products of one evolutionary 
process which swept across primordial matter and 
caused it to aggregate ; they were not formed individually 
by haphazard collisions having no particular time con- 
nection with one another. But the Phoenix complex is 
still active. Matter, we believe, is gradually destroyed 
and its energy set free in radiation. Is there no coun- 
ter-process by which radiation collects in space, evolves 
into electrons and protons, and begins star-building all 
over again? This is pure speculation and there is not 
much to be said on one; side or the other as to its truth. 
But I would mildly criticise the mental outlook which 
wishes it to be true. However much we eliminate the 
minor extravagances of Nature, we do not by these 
theories stop the inexorable running-down of the world 



86 THE RUNNING-DOWN OF THE UNIVERSE 

by loss of organisation and increase of the random 
element. Whoever wishes for a universe which can 
continue indefinitely in 'activity must lead a crusade 
against the second law of thermodynamics; the possi- 
bility of re-formation of matter from radiation is not 
crucial and we can await conclusions with some indif- 
ference. 

At present we can see no way in which an attack on 
the second law of thermodynamics could possibly 
succeed, and I confess that personally I have no great 
desire that it should succeed in averting the final 
running-down of the universe. I am no Phoenix 
worshipper. This is a topic on which science is silent, 
and all that one can say is prejudice. But since prejudice 
in favour of a never-ending cycle of rebirth of matter 
and worlds is often vocal, I may perhaps give voice to 
the opposite prejudice. I would feel more content that 
the universe should accomplish some great scheme of 
evolution and, having achieved whatever may be 
achieved, lapse back into chaotic changelessness, than 
that its purpose should be banalised by continual 
repetition. I am an Evolutionist, not a Multiplicationist. 
It seems rather stupid to keep doing the same thing 
over and over again. 



Chapter V 
"BECOMING" 

Linkage of Entropy with Becoming. When you say to 
yourself, "Every day I grow better and better", science 
churlishly replies — 

"I see no signs of it. I see you extended as a four- 
dimensional worm in space-time; and, although goodness 
is not strictly within my province, I will grant that one 
end of you is better than the other. But whether you 
grow better or worse depends on which way up I hold 
you. There is in your consciousness an idea of growth 
or 'becoming' which, if it is not illusory, implies that 
you have a label 'This side up'. I have searched for 
such a label all through the physical world and can find 
no trace of it, so I strongly suspect that the label is 
non-existent in the world of reality." 

That is the reply of science comprised in primary 
law. Taking account of secondary law, the reply is 
modified a little, though it is still none too gracious — 

"I have looked again and, in the course of studying 
a property called entropy, I find that the physical world 
is marked with an arrow which may possibly be in- 
tended to indicate which way up it should be regarded. 
With that orientation I find that you really do grow 
better. Or, to speak precisely, your good end is in the 
part of the world with most entropy and your bad end 
in the part with least. Why this arrangement should 
be considered more creditable than that of your neigh- 
bour who has his good and bad ends the other way 
round, I cannot imagine." 

A problem here rises before us concerning the 

87 



88 "BECOMING" 

linkage of the symbolic world of physics to the world 
of familiar experience. As explained in the Introduction 
this question of linkage remains over at the end of the 
strictly physical investigations. Our present problem 
is to understand the linkage between entropy which 
provides time's arrow in the symbolic world and the 
experience of growing or becoming which is the inter- 
pretation of time's arrow in the familiar world. We 
have, I think, shown exhaustively in the last chapter that 
the former is the only scientific counterpart to the latter. 

But in treating change of entropy as a symbolic 
equivalent for the moving on of time familiar to our 
minds a double difficulty arises. Firstly, the symbol 
seems to be of inappropriate nature; it is an elaborate 
mathematical construct, whereas we should expect so 
fundamental a conception as "becoming" to be among 
the elementary indefinables — the A B C of physics. 
Secondly, a symbol does not seem to be quite what is 
wanted; we want a significance which can scarcely be 
conveyed by a symbol of the customary metrical type — 
the recognition of a dynamic quality in external Nature. 
We do not "put sense into the world" merely by 
recognising that one end of it is more random than the 
other; we have to put a genuine significance of "be- 
coming" into it and not an artificial symbolic substitute. 

The linkage of entropy-change to "becoming" 
presents features unlike every other problem of paral- 
lelism of the scientific and familiar worlds. The usual 
relation is illustrated by the familiar perception of 
colour and its scientific equivalent electromagnetic wave- 
length. Here there is no question of resemblance 
between the underlying physical cause and the mental 
sensation which arises. All that we can require of 
the symbolic counterpart of colour is that it shall be 



ENTROPY AND BECOMING 89 

competent to pull the trigger of a (symbolic) nerve. The 
physiologist can trace the nerve mechanism up to the 
brain; but ultimately there is a hiatus which no one 
professes to fill up. Symbolically we may follow the 
influences of the physical world up to the door of the 
mind; they ring the door-bell and depart. 

But the association of "becoming" with entropy- 
change is not to be understood in the same way. It 
is clearly not sufficient that the change in the random 
element of the world should deliver an impulse at the 
end of a nerve, leaving the mind to create in response 
to this stimulus the fancy that it is turning the reel of 
a cinematograph. Unless we have been altogether 
misreading the significance of the world outside us — 
by interpreting it in terms of evolution and progress, 
instead of a static extension — we must regard the 
feeling of "becoming" as (in some respects at least) a 
true mental insight into the physical condition which 
determines it. It is true enough that whether we are 
dealing with the experience of "becoming" or with the 
more typical sense-experiences of light, sound, smell, 
etc., there must always be some point at which we lose 
sight of the physical entities ere they arise in new dress 
above our mental horizon. But if there is any experience 
in which this mystery of mental recognition can be 
interpreted as insight rather than image-building, it 
should be the experience of "becoming"; because in this 
case the elaborate nerve mechanism does not intervene. 
That which consciousness is reading off when it feels the 
passing moments lies just outside its door. Whereas, 
even if we had reason to regard our vivid impression 
of colour as insight, it could not be insight into the 
electric waves, for these terminate at the retina far from 
the seat of consciousness. 



9 o "BECOMING" 

I am afraid that the average reader will feel impa- 
tient with the long-winded discussion I am about to give 
concerning the dynamic character of the external world. 
"What is all the bother about? Why not make at once 
the hypothesis that 'becoming' is a kind of one-way 
texture involved fundamentally in the structure of 
Nature? The mind is cognisant of this texture (as it is 
cognisant of other features of the physical world) and 
apprehends it as the passing on of time — a fairly correct 
appreciation of its actual nature. As a result of this 
one-way texture the random element increases steadily 
in the direction of the grain, and thus conveniendy 
provides the physicist with an experimental criterion for 
determining the way of the grain; but it is the grain 
and not this particular consequence of it which is the 
direct physical counterpart of 'becoming'. It may be 
difficult to find a rigorous proof of this hypothesis; but 
after all we have generally to be content with hypotheses 
that rest only on plausibility." 

This is in fact the kind of idea which I wish to 
advocate; but the "average reader" has probably not 
appreciated that before the physicist can admit it, a 
delicate situation concerning the limits of scientific 
method and the underlying basis of physical law has to 
be faced. It is one thing to introduce a plausible 
hypothesis in order to explain observational phenomena; 
it is another thing to introduce it in order to give the 
world outside us a significant or purposive meaning, 
however strongly that meaning may be insisted on by 
something in our conscious nature. From the side of 
scientific investigation we recognise only the progressive 
change in the random element from the end of the world 
with least randomness to the end with most; that in itself 
gives no ground for suspecting any kind of dynamical 



ENTROPY AND BECOMING 91 

meaning. The view here advocated is tantamount to an 
admission that consciousness, looking out through a pri- 
vate door, can learn by direct insight an underlying char- 
acter of the world which physical measurements do not 
betray. 

In any attempt to bridge the domains of experience 
belonging to the spiritual and physical sides of our na- 
ture, Time occupies the key position. I have already re- 
ferred to its dual entry into our consciousness — through 
the sense organs which relate it to the other entities of 
the physical world, and directly through a kind of pri- 
vate door into the mind. The physicist, whose method 
of inquiry depends on sharpening up our sense organs by 
auxiliary apparatus of precision, naturally does not look 
kindly v on private doors, through which all formsl of 
superstitious fancy might enter unchecked. But is he 
ready to forgo that knowledge of the going on of time 
which has reached us through the door, and content 
himself with the time inferred from sense-impressions 
which is emaciated of all dynamic quality? 

No doubt some will reply that they are content; to 
these I would say — Then show your good faith by 
reversing the dynamic quality of time (which you may 
freely do if it has no importance in Nature), and, just 
for a change, give us a picture of the universe passing 
from the more random to the less random state, each 
step showing a gradual victory of antichance over 
chance. If you are a biologist, teach us how from Man 
and a myriad other primitive forms of life, Nature in 
the course of ages achieved the sublimely simple struc- 
ture of the amoeba. If you are an astronomer, tell how 
waves of light hurry in from the depths of space and 
condense on to the stars; how the complex solar system 
unwinds itself into the evenness of a nebula. Is this the 



92 "BECOMING" 

enlightened outlook which you wish to substitute for 
the first chapter of Genesis? If you genuinely believe 
that a contra-evolutionary theory is just as true and as 
significant as an evolutionary theory, surely it is time that 
a protest should be made against the entirely one-sided 
version currently taught. 



Dynamic Quality of the External World. But for our 
ulterior conviction of the dynamic quality of time, it 
would be possible to take the view that ''becoming" is 
purely subjective — that there is no "becoming" in the 
external world which lies passively spread out in the 
time-dimension as Minkowski pictured it. My con- 
sciousness then invents its own serial order for the sense 
impressions belonging to the different view-points along 
the track in the external world, occupied by the four- 
dimensional worm who is in some mysterious way 
Myself; and in focussing the sensations of a particular 
view-point I get the illusion that the corresponding 
external events are "taking place". I suppose that this 
would be adequate to account for the observed phe- 
nomena. The objections to it hinge on the fact that it 
leaves the external world without any dynamic quality 
intrinsic to it. 

It is useful to recognise how some of our most ele- 
mentary reasoning tacitly assumes the existence of this 
dynamic quality or trend; to eradicate it would almost 
paralyse our faculties of inference. In the operation of 
shuffling cards it seems axiomatic that the cards must 
be in greater disarrangement at a later instant. Can 
you conceive Nature to be such that this is not obviously 
true? But what do we here mean by "later"? So far 
as the axiomatic character of the conclusion is concerned 



DYNAMIC QUALITY OF THE WORLD 93 

(not its experimental verification) we cannot mean 
"later" as judged by consciousness; its obviousness is 
not bound up with any speculations as to the behaviour 
of consciousness. Do we then mean "later" as judged 
by the physical criterion of time's arrow, i.e. corre- 
sponding to a greater proportion of the random element? 
But that would be tautological — the cards are more 
disarranged when there is more of the random element. 
We did not mean a tautology; we unwittingly accepted 
as a basis for our thought about the question an unam- 
biguous trend from past to future in the space-time where 
the operation of shuffling is performed. 

The crux of the matter is that, although a change 
described as sorting is the exact opposite to a change 
described as shuffling we cannot imagine a cause of 
sorting to be the exact opposite of a cause of shuffling. 
Thus a reversal of the time-direction which turns 
shuffling into sorting does not make the appropriate 
transformation of their causes. Shuffling can have in- 
organic causes, but sorting is the prerogative of mind or 
instinct. We cannot believe that it is merely an orienta- 
tion with respect to the time-direction which differentiates 
us from inorganic nature. Shuffling is related to sorting 
(so far as the change of configuration is concerned) as 
plus is to minus; but to say that the cause of shuffling 
is related to the cause of sorting in the same way would 
seem equivalent to saying that the activities of matter 
and mind are related like plus and minus — which 
surely is nonsense. Hence if we view the world from 
future to past so that shuffling and sorting are inter- 
changed, their causes do not follow suit, and the rational 
connection is broken. To restore coherency we must 
postulate that by this change of direction something 
else has been reversed, viz. the trend in world-texture 



94 "BECOMING" 

spoken of above; "becoming" has been turned into 
"unbecoming". If we like we can now go on to account, 
not for things becoming unshuffled, but for their un- 
becoming shuffled — and, if we wish to pursue this aspect 
further, we must discuss not the causes but the un- 
causes. But, without tying ourselves into verbal knots, 
the meaning evidently is that "becoming" gives a 
texture to the world which it is illegitimate to reverse. 

Objectivity of Becoming. In general we should describe 
the familiar world as subjective and the scientific world 
as objective. Take for instance our former example of 
parallelism, viz. colour in the familiar world and its 
counterpart electromagnetic wave-length in the scientific 
world. Here we have little hesitation in describing the 
waves as objective and the colour as subjective. The 
wave is the reality — or the nearest we can get to a 
description of reality; the colour is mere mind-spinning. 
The beautiful hues which flood our consciousness under 
stimulation of the waves have no relevance to the ob- 
jective reality. For a colour-blind person the hues are 
different; and although persons of normal sight make 
the same distinctions of colour, we cannot ascertain 
whether their consciousness of red, blue, etc. is just like 
our own. Moreover, we recognise that the longer and 
shorter electromagnetic waves which have no visual 
effect associated with them are just as real as the col- 
oured waves. In this and other parallelisms we find the 
objective in the scientific world and the subjective in the 
familiar world. 

But in the parallelism between entropy-gradient and 
"becoming" the subjective and objective seem to have 
got on to the wrong sides. Surely "becoming" is a 
reality — or the nearest we can get to a description of 



OBJECTIVITY OF BECOMING 95 

reality. We are convinced that a dynamic character 
must be attributed to the external world; making all 
allowance for mental imagery, I do not see how the 
essence of "becoming" can be much different from 
what it appears to us to be. On the other side we have 
entropy which is frankly of a much more subjective 
nature than most of the ordinary physical qualities. 
Entropy is an appreciation of arrangement and or- 
ganisation; it is subjective in the same sense that the 
constellation Orion is subjective. That which is arranged 
is objective, so too are the stars composing the con- 
stellation; but the association is the contribution of the 
mind which surveys. If colour is mind-spinning, so also 
is entropy a mind-spinning — of the statistician. It has 
about as much objectivity as a batting average. 

Whilst the physicist would generally say that the 
matter of this familiar table is really a curvature of 
space, and its colour is really electromagnetic wave- 
length, I do not think he would say that the familiar 
moving on of time is really an entropy-gradient. I am 
quoting a rather loose way of speaking; but it reveals 
that there is a distinct difference in our attitude towards 
the last parallelism. Having convinced ourselves that 
the two things are connected, we must conclude that 
there is something as yet ungrasped behind the notion 
of entropy — some mystic interpretation, if you like — 
which is not apparent in the definition by which we 
introduced it into physics. " In short we strive to see 
that entropy-gradient may really be the moving on of 
time (instead of vice versa). 

Before passing on I would note that this exceptional 
appearance of subjective and objective apparently in 
their wrong worlds gives food for thought. It may 
prepare us for a view of the scientific world adopted in 



96 "BECOMING" 

the later chapters which is much more subjective than 
that usually held by science. 

The more closely we examine the association of 
entropy with "becoming" the greater do the obstacles 
appear. If entropy were one of the elementary in- 
definables of physics there would be no difficulty. Or 
if the moving on of time were something of which we 
were made aware through our sense organs there would 
be no difficulty. But the actual combination which we 
have to face seems to be unique in its difficulty. 

Suppose that we had had to identify "becoming" 
with electrical potential-gradient instead of with en- 
tropy-change. We discover potential through the 
readings of a voltmeter. The numerical reading stands 
for something in the condition of the world, but we form 
no picture of what that something is. In scientific 
researches we only make use of the numerical value — 
a code-number attached to a background outside all 
conception. It would be very interesting if we could 
relate this mysterious potential to any of our familiar 
conceptions. Clearly, if we could identify the change 
of potential with the familiar moving on of time, we 
should have made a great step towards grasping its 
intrinsic nature. But turning from supposition to fact, 
we have to identify potential-gradient with force. Now 
it is true that we have a familiar conception of force — 
a sensation of muscular effort. But this does not give 
us any idea of the intrinsic nature of potential-gradient; 
the sensation is mere mind-spinning provoked by 
nervous impulses which have travelled a long way from 
the seat of the force. That is the way with all physical 
entities which affect the mind through the sense organs. 
The interposed nerve-mechanism would prevent any close 
association of the mental image with the physical cause, 



OBJECTIVITY OF BECOMING 97 

even if we were disposed to trust our mental insight when 
it has a chance of operating directly. 

Or suppose that we had had to identify force with 
entropy-gradient. That would only mean that entropy- 
gradient is a condition which stimulates a nerve, which 
thereupon transmits an impulse to the brain, out of 
which the mind weaves its own peculiar impression of 
force. No one would feel intuitive objection to the 
hypothesis that the muscular sensation of force is 
associated with change of organisation of the molecules 
of the muscle. 

Our trouble is that we have to associate two things, 
both of which we more or less understand, and, so far 
as we understand them, they are utterly different. It 
is absurd to pretend that we are in ignorance of the 
nature of organisation in the external world in the same 
way that we are ignorant of the intrinsic nature of 
potential. It is absurd to pretend that we have no 
justifiable conception of "becoming" in the external 
world. That dynamic quality — that significance which 
makes a development from past to future reasonable 
and a development from future to past farcical — has to 
do much more than pull the trigger of a nerve. It is so 
welded into our consciousness that a moving on of 
time is a condition of consciousness. We have direct 
insight into "becoming" which sweeps aside all sym- 
bolic knowledge as on an inferior plane. If I grasp the 
notion of existence because I myself exist, I grasp the no- 
tion of becoming because I myself become. It is the in- 
nermost Ego of all which is and becomes. 

The incongruity of symbolising this fundamental 
intuition by a property of arrangement of the micro- 
scopic constituents of the world, is evident. What this 
difficulty portends is still very obscure. But it is not 



9 8 "BECOMING" 

irrelevant to certain signs of change which we may 
discern in responsible scientific opinion with regard to 
the question of primary and secondary law. The cast- 
iron determinism of primary law is, I think, still widely 
accepted but no longer unquestioningly. It now seems 
clear that we have not yet got hold of any primary law 
— that all those laws at one time supposed to be primary 
are in reality statistical. No doubt it will be said that 
that was only to be expected; we must be prepared for 
a very long search before we get down to ultimate 
foundations, and not be disappointed if new discoveries 
reveal unsuspected depths beneath. But I think it might 
be said that Nature has been caught using rather unfair 
dodges to prevent our discovering primary law — that 
kind of artfulness which frustrated our efforts to discover 
velocity relative to the aether.* I believe that Nature is 
honest at heart, and that she only resorts to these ap- 
parent shifts of concealment when we are looking for 
something which is not there. It is difficult to see now 
any justification for the strongly rooted conviction in the 
ultimate re-establishment of a deterministic scheme of 
law except a supposed necessity of thought. Thought 
has grown accustomed to doing without a great many 
"necessities" in recent years. 

One would not be surprised if in the reconstruction 
of the scheme of physics which the quantum theory is 
now pressing on us, secondary law becomes the basis 
and primary law is discarded. In the reconstructed 
world nothing is impossible though many things are 
improbable. The effect is much the same, but the kind 
of machinery that we must conceive is altogether 
different. We shall have further glimpses of this problem 
and I will not here pursue it. Entropy, being a quantity 

* See p. 23i. 



OUR DUAL RECOGNITION OF TIME 99 

introduced in connection with secondary law will now 
exist, so to speak, in its own right instead of by its 
current representation as arrangement of the quantities 
in the abandoned primary scheme; and in that right it 
may be more easily accepted as the symbol for the 
dynamic quality of the world. I cannot make my 
meaning more precise, because I am speaking of a 
still hypothetical change of ideas which no one has been 
able to bring about. 

Our Dual Recognition of Time, Another curiosity which 
strikes us is the divorce in physics between time and 
time's arrow. A being from another world who wishes 
to discover the temporal relation of two events in this 
world has to read two different indicators. He must 
read a clock in order to find out how much later one 
event is than the other, and he must read some arrange- 
ment for measuring the disorganisation of energy (e.g. a 
thermometer) in order to discover which event is the 
later.* The division of labour is especially striking 
when we remember that our best clocks are those in 
which all processes such as friction, which introduce 
disorganisation of energy, are eliminated as far as 
possible. The more perfect the instrument as a meas- 
urer of time, the more completely does it conceal time's 
arrow. 



* To make the test strictly from another world he must not assume 
that the figures marked on the clock-dial necessarily go the right way 
round; nor must he assume that the progress of his consciousness has 
any relation to the flow of time in our world. He has, therefore, merely 
two dial-readings for the two events without knowing whether the 
difference should be reckoned plus or minus. The thermometer would 
be used in conjunction with a hot and cold body in contact. The differ- 
ence of the thermometer readings for the two bodies would be taken at 
the moment of each event. The event for which the difference is smaller 
is the later. 



ioo "BECOMING" 

This paradox seems to be explained by the fact 
pointed out in chapter in that time comes into our 
consciousness by two routes. We picture the mind like 
an editor in his sanctum receiving through the nerves 
scrappy messages from all over the outside world, and 
making a story of them with, I fear, a good deal of 
editorial invention. Like other physical quantities time 
enters in that way as a particular measurable relation 
between events in the outside world; but it comes in 
without its arrow. In addition our editor himself ex- 
periences a time in his consciousness — the temporal 
relation along his own track through the world. This 
experience is immediate, not a message from outside, 
but the editor realises that what he is experiencing is 
equivalent to the time described in the messages. Now 
consciousness declares that this private time possesses 
an arrow, and so gives a hint to search further for the 
missing arrow among the messages. The curious thing 
is that, although the arrow is ultimately found among 
the messages from outside, it is not found in the mes- 
sages from clocks, but in messages from thermometers 
and the like instruments which do not ordinarily pretend 
to measure time. 

Consciousness, besides detecting time's arrow, also 
roughly measures the passage of time. It has the right 
idea of time-measurement, but is a bit of a bungler in 
carrying it out. Our consciousness somehow manages 
to keep in close touch with the material world, and we 
must suppose that its record of the flight of time is the 
reading of some kind of a clock in the material of the 
brain — possibly a clock which is a rather bad time- 
keeper. I have generally had in mind in this connection 
an analogy with the clocks of physics designed for good 
time-keeping; but I am now inclined to think that a 



OUR DUAL RECOGNITION OF TIME 101 

better analogy would be an entropy-clock, i.e. an in- 
strument designed primarily for measuring the rate of 
disorganisation of energy, and only very roughly keep- 
ing pace with time. 

A typical entropy-clock might be designed as follows. 
An electric circuit is composed of two different metals 
with their two junctions embedded respectively in a 
hot and cold body in contact. The circuit contains a 
galvanometer which constitutes the dial of the entropy- 
clock. The thermoelectric current in the circuit is 
proportional to the difference of temperature of the two 
bodies; so that as the shuffling of energy between them 
proceeds, the temperature difference decreases and the 
galvanometer reading continually decreases. This clock 
will infallibly tell an observer from another world which 
of two events is the later. We have seen that no ordi- 
nary clock can do this. As to its time-keeping qualities 
we can only say that the motion of the galvanometer 
needle has some connection with the rate of passage of 
time — which is perhaps as much as can be said for the 
time-keeping qualities of consciousness. 

It seems to me, therefore, that consciousness with its 
insistence on time's arrow and its rather erratic ideas of 
time measurement may be guided by entropy-clocks in 
some portion of the brain. That avoids the unnatural 
assumption that we consult two different cells of the 
material brain in forming our ideas of duration and of 
becoming, respectively. Entropy-gradient is then the 
direct equivalent of the time of consciousness in both 
its aspects. Duration measured by physical clocks (time- 
like interval) is only remotely connected. 

Let us try to clear up our ideas of time by a summary 
of the position now reached. Firstly, physical time is a 



102 "BECOMING" 

system of partitions in the four-dimensional world 
(world-wide instants). These are artificial and relative 
and by no means correspond to anything indicated to 
us by the time of consciousness. Secondly, we recognise 
in the relativity theory something called a temporal 
relation which is absolutely distinct from a spatial 
relation. One consequence of this distinction is that the 
mind attached to a material body can only traverse a 
temporal relation; so that, even if there is no closer 
connection, there is at least a one-to-one correspondence 
between the sequence of phases of the mind and a 
sequence of points in temporal relation. Since the mind 
interprets its own sequence as a time of consciousness, we 
can at least say that the temporal relation in physics 
has a connection with the time of consciousness which 
the spatial relation does not possess. I doubt if the 
connection is any closer. I do not think the mental 
sequence is a "reading off" of the physical temporal 
relation, because in physics the temporal relation is 
arrowless. I think it is a reading off of the physical 
entropy-gradient, since this has the necessary arrow. 
Temporal relation and entropy-gradient, both rigorously 
defined in physics, are entirely distinct and in general 
are not numerically related. But, of course, other things 
besides time can "keep time"; and there is no reason 
why the generation of the random element in a special 
locality of the brain should not proceed fairly uniformly. 
In that case there will not be too great a divergence 
between the passage of time in consciousness and the 
length of the corresponding temporal relation in the 
physical world. 



THE REACTION FROM ANALYSIS 103 

The Scientific Reaction from Microscopic Analysis. From 
the point of view of philosophy of science the con- 
ception associated with entropy must I think be ranked 
as the great contribution of the nineteenth century to 
scientific thought. It marked a reaction from the view 
that everything to which science need pay attention is 
discovered by a microscopic dissection of objects. It 
provided an alternative standpoint in which the centre 
of interest is shifted from the entities reached by the 
customary analysis (atoms, electric potentials, etc.) to 
qualities possessed by the system as a whole, which 
cannot be split up and located — a little bit here, and a 
little bit there. The artist desires to convey significances 
which cannot be told by microscopic detail and accord- 
ingly he resorts to impressionist painting. Strangely 
enough the physicist has found the same necessity; but 
his impressionist scheme is just as much exact science 
and even more practical in its application than his micro- 
scopic scheme. 

Thus in the study of the falling stone the microscopic 
analysis reveals myriads of separate molecules. The 
energy of the stone is distributed among the molecules, 
the sum of the energies of the molecules making up the 
energy of the stone. But we cannot distribute in that 
way the organisation or the random element in the 
motions. It would be meaningless to say that a particu- 
lar fraction of the organisation is located in a par- 
ticular molecule. 

There is one ideal of survey which would look into 
each minute compartment of space in turn to see what 
it may contain and so make what it would regard as 
a complete inventory of the world. But this misses 
any world-features which are not located in minute 
compartments. We often think that when we have 



104 "BECOMING" 

completed our study of one we know all about two> be- 
cause "two" is "one and one". We forget that we have 
still to make a study of "and". Secondary physics is 
the study of "and" — that is to say, of organisation. 

Thanks to clear-sighted pioneers in the last century 
science became aware that it was missing something of 
practical importance by following the inventory method 
of the primary scheme of physics. Entropy became 
recognised although it was not found in any of the com- 
partments. It was discovered and exalted because it was 
essential to practical applications of physics, not to 
satisfy any philosophic hungering. But by it science 
has been saved from a fatal narrowness. If we had kept 
entirely to the inventory method, there would have been 
nothing to represent "becoming" in the physical world. 
And science, having searched high and low, would 
doubtless have reported that "becoming" is an un- 
founded mental illusion — like beauty, life, the soul, and 
other things which it is unable to inventory. 

I think that doubts might well have been entertained 
as to whether the newcomer was strictly scientific. 
Entropy was not in the same category as the other 
physical quantities recognised in science, and the ex- 
tension — as we shall presently see — was in a very 
dangerous direction. Once you admit attributes of 
arrangement as subject-matter of physics, it is difficult 
to draw the line. But entropy had secured a firm place 
in physics before it was discovered that it was a measure 
of the random element in arrangement. It was in great 
favour with the engineers. Their sponsorship was the 
highest testimonial to its good character; because at that 
time it was the general assumption that the Creation was 
the work of an engineer (not of a mathematician, as is 
the fashion nowadays). 



THE REACTION FROM ANALYSIS 105 

Suppose that we were asked to arrange the following 
in two categories — 

distance, mass, electric force, entropy, beauty, melody. 

I think there are the strongest grounds for placing 
entropy alongside beauty and melody and not with the 
first three. Entropy is only found when the parts are 
viewed in association, and it is by viewing or hearing 
the parts in association that beauty and melody are 
discerned. All three are features of arrangement. It is 
a pregnant thought that one of these three associates 
should be able to figure as a commonplace quantity of 
science. The reason why this stranger can pass itself 
off among the aborigines of the physical world is, that 
it is able to speak their language, viz. the language of 
arithmetic. It has a measure-number associated with it 
and so is made quite at home in physics. Beauty and 
melody have not the arithmetical pass-word and so are 
barred out. This teaches us that what exact science looks 
out for is not entities of some particular category, but 
entities with a metrical aspect. We shall see in a later 
chapter that when science admits them it really admits 
only their metrical aspect and occupies itself solely with 
that. It would be no use for beauty, say, to fake up a 
few numerical attributes (expressing for instance the 
ideal proportions of symmetry) in the hope of thereby 
gaining admission into the portals of science and carrying 
on an aesthetic crusade within. It would find that the 
numerical aspects were duly admitted, but the aesthetic 
significance of them left outside. So also entropy is 
admitted in its numerical aspect; if it has as we faintly 
suspect some deeper significance touching that which 
appears in our consciousness as purpose (opposed to 
chance) , that significance is left outside. These fare no 



106 "BECOMING" 

worse than mass, distance, and the like which surely 
must have some significance beyond mere numbers; if 
so, that significance is lost on their incorporation into 
the scientific scheme — the world of shadows. 

You may be inclined to regard my insistence that 
entropy is something excluded from the inventory of 
microscopic contents of the world as word-splitting. If 
you have all the individuals before you, their associations, 
arrangement and organisation are automatically before 
you. If you have the stars, you have the constellations. 
Yes; but if you have the stars, you do not take the 
constellations seriously. It had become the regular 
outlook of science, closely associated with its materialistic 
tendencies, that constellations are not to be taken 
seriously, until the constellation of entropy made a 
solitary exception. When we analyse the picture into 
a large number of particles of paint, we lose the aes- 
thetic significance of the picture. The particles of paint 
go into the scientific inventory, and it is claimed that 
everything that there really was in the picture is kept. 
But this way of keeping a thing may be much the same 
as losing it. The essence of a picture (as distinct from 
the paint) is arrangement. Is arrangement kept or lost? 
The current answer seems inconsistent. In so far as 
arrangement signifies a picture, it is lost; science has 
to do with paint, not pictures. In so far as arrangement 
signifies organisation it is kept; science has much to do 
with organisation. Why should we (speaking now as 
philosophers, not scientists) make a discrimination 
between these two aspects of arrangement? The dis- 
crimination is made because the picture is no use to the 
scientist — he cannot get further with it. As impartial 
judges it is our duty to point out that likewise entropy 
is no use to the artist — he cannot develop his outlook 
with it. 



INSUFFICIENCY OF PRIMARY LAW 107 

I am not trying to argue that there is in the external 
world an objective entity which is the picture as distinct 
from the myriads of particles into which science has 
analysed it. I doubt if the statement has any meaning; 
nor, if it were true, would it particularly enhance my 
esteem of the picture. What I would say is this: 
There is a side of our personality which impels us to 
dwell on beauty and other aesthetic significances in 
Nature, and in the work of man, so that our environ- 
ment means to us much that is not warranted by any- 
thing found in the scientific inventory of its struc- 
ture. An overwhelming feeling tells us that this is 
right and indispensable to the purpose of our existence. 
But is it rational? How can reason regard it otherwise 
than as a perverse misrepresentation of what is after all 
only a collection of atoms, aether-waves and the like, 
going about their business? If the physicist as advocate 
for reason takes this line, just whisper to him the word 
Entropy. 

Insufficiency of Primary Law. I daresay many of my 
physical colleagues will join issue with me over the 
status I have allowed to entropy as something foreign 
to the microscopic scheme, but essential to the physical 
world. They would regard it rather as a labour-saving 
device, useful but not indispensable. Given any practical 
problem ordinarily solved by introducing the conception 
of entropy, precisely the same result could be reached 
(more laboriously) by following out the motion of each 
individual particle of matter or quantum of energy under 
the primary microscopic laws without any reference to 
entropy explicit or implicit. Very well ; let us try. There's 
a problem for you — 

[A piece of chalk was thrown on the lecture table 
where it rolled and broke into two pieces.] 



108 "BECOMING" 

You are given the instantaneous position and velocity* 
of every molecule, or if you like every proton and 
electron, in those pieces of chalk and in as much of the 
table and surrounding air as concerns you. Details of 
the instantaneous state of every element of energy are 
also given. By the microscopic (primary) laws of mo- 
tion you can trace the state from instant to instant. 
You can trace how the atoms moving aimlessly within 
the lumps of chalk gradually form a conspiracy so that 
the lumps begin to move as a whole. The lumps bounce 
a little and roll on the table; they come together and 
join up; then the whole piece of chalk rises gracefully 
in the air, describes a parabola, and comes to rest be- 
tween my fingers. I grant that you can do all that with- 
out requiring entropy or anything outside the limits of 
microscopic physics. You have solved the problem. 
But, have you quite got hold of the significance of your 
solution? Is it quite a negligible point that what you 
have described from your calculations is an unhappen- 
ingf There is no need to alter a word of your descrip- 
tion so far as it goes; but it does seem to need an 
addendum which would discriminate between a trick 
worthy of Mr. Maskelyne and an ordinary everyday 
unoccurrence. 

The physicist may say that the addendum asked for 
relates to significance, and he has nothing to do with 
significances; he is only concerned that his calculations 
shall agree with observation. He cannot tell me whether 
the phenomenon has the significance of a happening or 
an unhappening; but if a clock is included in the 

* Velocities are relative to a frame of space and time. Indicate which 
frame you prefer, and you will be given velocity relative to that frame. 
(This throws on you the responsibility for any labelling of the frame — 
left, right, past future* etc.) 



INSUFFICIENCY OF PRIMARY LAW 109 

problem he can give the readings of the clock at each 
stage. There is much to be said for excluding the whole 
field of significance from physics; it is a healthy reaction 
against mixing up with our calculations mystic con- 
ceptions that (officially) we know nothing about. 
I rather envy the pure physicist his impregnable position. 
But if he rules significances entirely outside his scope, 
somebody has the job of discovering whether the physi- 
cal world of atoms, aether and electrons has any signifi- 
cance whatever. Unfortunately for me I am expected in 
these lectures to say how the plain man ought to regard 
the scientific world when it comes into competition with 
other views of our environment. Some of my audience 
may not be interested in a world invented as a mere 
calculating device. Am I to tell them that the scientific 
world has no claim on their consideration when the eter- 
nal question surges in the mind, What is it all about? I 
am sure my physical colleagues will expect me to put up 
some defence of the scientific world in this connection. 
I am ready to do so; only I must insist as a preliminary 
that we should settle which is the right way up of it. 
I cannot read any significance into a physical world 
when it is held before me upside down, as happened 
just now. For that reason I am interested in entropy 
not only because it shortens calculations which can be 
made by other methods, but because it determines an 
orientation which cannot be found by other methods. 

The scientific world is, as I have often repeated, a 
shadow-world, shadowing a world familiar to our con- 
sciousness. Just how much do we expect it to shadow? 
We do not expect it to shadow all that is in our mind, 
emotions, memory, etc. In the main we expect it to 
shadow impressions which can be traced to external 
sense-organs. But time makes a dual entry and thus 



no "BECOMING" 

t 

forms an intermediate link between the internal and the 
external. This is shadowed partially by the scientific 
world of primary physics (which excludes time's ar- 
row), but fully when we enlarge the scheme to include 
entropy. Therefore by the momentous departure in the 
nineteenth century the scientific world is not confined to 
a static extension around which the mind may spin a 
romance of activity and evolution; it shadows that 
dynamic quality of the familiar world which cannot be 
parted from it without disaster to its significance. 

In sorting out the confused data of our experience it 
has generally been assumed that the object of the quest 
is to find out all that really exists. There is another 
quest not less appropriate to the nature of our experience 
— to find out all that really becomes. 



Chapter VI 
GRAVITATION— THE LAW 

You sometimes speak of gravity as essential and inherent to matter. Pray 
do not ascribe that notion to me; for the cause of gravity is what I do not 
pretend to know, and therefore would take more time to consider of 
it. . . . 

Gravity must be caused by some agent acting constantly according 
to certain laws; but whether this agent be material or immaterial I have 
left to the consideration of my readers. 

Newton, Letters to Bentley. 

The Man in the Lift. About 19 15 Einstein made a 
further development of his theory of relativity extending 
it to non-uniform motion. The easiest way to approach 
this subject is by considering the Man in the Lift. 

Suppose that this room is a lift. The support breaks 
and down we go with ever-increasing velocity, falling 
freely. 

Let us pass the time by performing physical experi- 
ments. The lift is our laboratory and we shall start at 
the beginning and try to discover all the laws of Nature 
— that is to say, Nature as interpreted by the Man in 
the Lift. To a considerable extent this will be a repeti- 
tion of the history of scientific discovery already made 
in the laboratories on terra firma. But there is one 
notable difference. 

I perform the experiment of dropping an apple held 
in the hand. The apple cannot fall any more than it 
was doing already. You remember that our lift and all 
things contained in it are falling freely. Consequently 
the apple remains poised 'by my hand. There is one 
incident in the history of science which will not repeat 
itself to the men in the lift, viz. Newton and the apple 
tree. The magnificent conception that the agent which 

in 



ii2 GRAVITATION— THE LAW 

guides the stars in their courses is the same as that 
which in our common experience causes apples to drop, 
breaks down because it is our common experience in the 
lift that apples do not drop. 

I think we have now sufficient evidence to prove that 
in all other respects the scientific laws determined in 
the lift will agree with those determined under more 
orthodox conditions. But for this one omission the men 
in the lift will derive all the laws of Nature with which 
w r e are acquainted, and derive them in the same form 
that we have derived them. Only the force which 
causes apples to fall is not present in their scheme. 

I am crediting our observers in the lift with the usuai 
egocentric attitude, viz. the aspect of the world to me 
is its natural one. It does not strike them as odd to 
spend their lives falling in a lift; they think it much 
more odd to be perched on the earth's surface. There- 
fore although they perhaps have calculated that to beings 
supported in this strange way apples would seem to 
have a perplexing habit of falling, they do not take our 
experience of the ways of apples any more seriously 
than we have hitherto taken theirs. 

Are we to take their experience seriously? Or to put 
it another way — What is the comparative importance to 
be attached to a scheme of natural laws worked out by 
observers in the falling lift and one worked out by 
observers on terra ftrmal Is one truer than the other? 
Is one superior to the other? Clearly the difference if 
any arises from the fact that the schemes are referred 
to different frames of space and time. Our frame is a 
frame in which the solid ground is at rest; similarly their 
frame is a frame in which their lift is at rest. We have 
had examples before of observers using different frames, 
but those frames differed by a uniform velocity. The 



THE MAN IN THE LIFT 113 

velocity of the lift is ever-increasing — accelerated. Can 
we extend to accelerated frames our principle that 
Nature is indifferent to frames of space and time, so 
that no one frame is superior to any other? I think we 
can. The only doubt that arises is whether we should 
not regard the frame of the man in the lift as superior 
to, instead of being merely coequal with, our usual 
frame. 

When we stand on the ground the molecules of the 
ground support us by hammering on the soles of our 
boots with force equivalent to some ten stone weight. 
But for this we should sink through the interstices of 
the floor. We are being continuously and vigorously 
buffeted. Now this can scarcely be regarded as the ideal 
condition for a judicial contemplation of our natural 
surroundings, and it would not be surprising if our 
senses suffering from this treatment gave a jaundiced 
view of the world. Our bodies are to be regarded as 
scientific instruments used to survey the world. We 
should not willingly allow anyone to hammer on a 
galvanometer when it was being used for observation; 
and similarly it is preferable to avoid a hammering on 
one's body when it is being used as a channel of scien- 
tific knowledge. We get rid of this hammering when 
we cease to be supported. 

Let us then take a leap over a precipice so that we 
may contemplate Nature undisturbed. Or if that seems 
to you an odd way of convincing yourself that bodies do 
not fall,* let us enter the runaway lift again. Here 
nothing need be supported; our bodies, our galvano- 

* So far as I can tell (without experimental trial) the man who jumped 
over a precipice would soon lose all conception of falling; he would only 
notice that the surrounding objects were impelled past him with ever- 
increasing speed. 



ii 4 GRAVITATION— THE LAW 

meters, and all measuring apparatus are relieved of 
hammering and their indications can be received without 
misgiving. The space- and time-frame of the falling lift 
is the frame natural to observers who are unsupported; 
and the laws of Nature determined in these favourable 
circumstances should at least have not inferior status to 
those established by reference to other frames. 

I perform another experiment. This time I take two 
apples and drop them at opposite ends of the lift. What 
will happen? Nothing much at first; the apples remain 
poised where they were let go. But let us step outside 
the lift for a moment to watch the experiment. The two 
apples are pulled by gravity towards the centre of the 
earth. As they approach the centre their paths con- 
verge and they will meet at the centre. Now step back 
into the lift again. To a first approximation the apples 
remain poised above the floor of the lift; but presently 
we notice that they are drifting towards one another, 
and they will meet at the moment when (according to 
an outside observer) the lift is passing through the 
centre of the earth. Even though apples (in the lift) 
do not tend to fall to the floor there is still a mystery 
about their behaviour; and the Newton of the lift may 
yet find that the agent which guides the stars in their 
courses is to be identified with the agent which plays 
these tricks with apples nearer home. 

It comes to this. There are both relative and absolute 
features about gravitation. The feature that impresses 
us most is relative — relative to a frame that has no 
special importance apart from the fact that it is the one 
commonly used by us. This feature disappears alto- 
gether in the frame of the man in the lift and we ought 
to disregard it in any attempt to form an absolute pic- 
ture of gravitation. But there always remains something 



A NEW PICTURE OF GRAVITATION 115 

absolute, of which we must try to devise an appropriate 
picture. For reasons which I shall presently explain we 
find that it can be pictured as a curvature of space and 
time. 

A New Picture of Gravitation. The Newtonian picture 
of gravitation is a tug applied to the body whose path 
is disturbed. I want to explain why this picture must 
be superseded. I must refer again to the famous incident 
in which Newton and the apple-tree were concerned. 
The classical conception of gravitation is based on New- 
ton's account of what happened; but it is time to hear 
what the apple had to say. The apple with the usual 
egotism of an observer deemed itself to be at rest; 
looking down it saw the various terrestrial objects includ- 
ing Newton rushing upwards with accelerated velocity 
to meet it. Does it invent a mysterious agency or tug 
to account for their conduct? No; it points out that 
the cause of their acceleration is quite evident. Newton 
is being hammered by the molecules of the ground 
underneath him. This hammering is absolute — no ques- 
tion of frames of reference. With a powerful enough 
magnifying appliance anyone can see the molecules at 
work and count their blows. According to Newton's 
own law of motion this must give him an acceleration, 
which is precisely what the apple has observed. New- 
ton had to postulate a mysterious invisible force pulling 
the apple down; the apple can point to an evident cause 
propelling Newton up. 

The case for the apple's view is so overwhelming that 
I must modify the situation a little in order to give 
Newton a fair chance; because I believe the apple is 
making too much of a merely accidental advantage. I 
will place Newton at the centre of the earth where 



u6 GRAVITATION— THE LAW 

gravity vanishes, so that he can remain at rest without 
support — without hammering. He looks up and sees 
apples falling at the surface of the earth, and as before 
ascribes this to a mysterious tug which he calls gravita- 
tion. The apple looks down and sees Newton approach- 
ing it; but this time it cannot attribute Newton's accelera- 
tion to any evident hammering. It also has to invent 
a mysterious tug acting on Newton. 

We have two frames of reference. In one of them 
Newton is at rest and the apple is accelerated; in the 
other the apple is at rest and Newton accelerated. In 
neither case is there a visible cause for the acceleration; 
in neither is the object disturbed by extraneous ham- 
mering. The reciprocity is perfect and there is no ground 
for preferring one frame rather than the other. We 
must devise a picture of the disturbing agent which will 
not favour one frame rather than the other. In this 
impartial humour a tug will not suit us, because if we 
attach it to the apple we are favouring Newton's frame 
and if we attach it to Newton we are favouring the 
apple's frame.* The essence or absolute part of gravi- 
tation cannot be a force on a body, because we are en- 
tirely vague as to the body to which it is applied. We 
must picture it differently. 

* It will probably be objected that since the phenomena here dis- 
cussed are evidently associated with the existence of a massive body (the 
earth), and since Newton makes his tugs occur symmetrically about that 
body whereas the apple makes its tugs occur unsymmetrically (vanishing 
where the apple is, but strong at the antipodes), therefore Newton's 
frame is clearly to be preferred. It would be necessary to go deeply into 
the theory to explain fully why we do not regard this symmetry as of 
first importance ; we can only say here that the criterion of symmetry 
proves to be insufficient to pick out a unique frame and does not draw 
a sharp dividing line between the frames that it would admit and those 
it would have us reject. After all we can appreciate that certain frames 
are more symmetrical than others without insisting on calling the sym- 
metrical ones '"right"' and unsymmetrical ones 'wrong'. 



A NEW PICTURE OF GRAVITATION 117 

The ancients believed that the earth was flat. The 
small part which they had explored could be represented 
without serious distortion on a flat map. When new 
countries were discovered it would be natural to think 
that they could be added on to the flat map. A familiar 
example of such a flat map is Mercator's projection, and 
you will remember that in it the size of Greenland 
appears absurdly exaggerated. (In other projections 
directions are badly distorted.) Now those who adhered 
to the flat-earth theory must suppose that the map gives 
the true size of Greenland — that the distances shown in 
the map are the true distances. How then w r ould they 
explain that travellers in that country reported that the 
distances seemed to be much shorter than they "really" 
were ? They would, I suppose, invent a theory that there 
was a demon living in Greenland who helped travellers 
on their way. Of course no scientist would use so crude 
a word; he would invent a Graeco-Latin polysyllable to 
denote the mysterious agent which made the journeys 
seem so short; but that is only camouflage. But now 
suppose the inhabitants of Greenland have developed 
their own geography. They find that the most important 
part of the earth's surface (Greenland) can be repre- 
sented without serious distortion on a flat map. But 
when they put in distant countries such as Greece the 
size must be exaggerated; or, as they would put it, there 
is a demon active in Greece who makes the journeys 
there seem different from what the flat map clearly 
shows them to be. The demon is never where you are; 
it is always the other fellow who is haunted by him. 
We now understand that the true explanation is that the 
earth is curved, and the apparent activities of the demon 
arise from forcing the curved surface into a flat map 
and so distorting the simplicity of things. 



u8 GRAVITATION— THE LAW 

What has happened to the theory of the earth has 
happened also to the theory of the world of space-time. 
An observer at rest at the earth's centre represents what 
is happening in a frame of space and time constructed 
on the usual conventional principles which give what 
is called a flat space-time. He can locate the events in 
his neighbourhood without distorting their natural sim- 
plicity. Objects at rest remain at rest; objects in uni- 
form motion remain in uniform motion unless there is 
some evident cause of disturbance such as hammering; 
light travels in straight lines. He extends this flat frame 
to the surface of the earth where he encounters the 
phenomenon of falling apples. This new phenomenon has 
to be accounted for by an intangible agency or demon 
called gravitation which persuades the apples to deviate 
from their proper uniform motion. But we can also start 
with the frame of the falling apple or of the man in the 
lift. In the lift-frame bodies at rest remain at rest; bodies 
in uniform motion remain in uniform motion. But, as we 
have seen, even at the corners of the lift this simplicity 
begins to fail; and looking further afield, say to the 
centre of the earth, it is necessary to postulate the acti- 
vity of a demon urging unsupported bodies upwards 
(relatively to the lift-frame). As we change from one 
observer to another — from one flat space-time frame to 
another — the scene of activity of the demon shifts. It 
is never where our observer is, but always away yonder. 
Is not the solution now apparent? The demon is sim- 
ply the complication which arises when we try to fit a 
curved world into a flat frame. In referring the world 
to a flat frame of space-time we distort it so that the 
phenomena do not appear in their original simplicity. 
Admit a curvature of the world and the mysterious 
agency disappears. Einstein has exorcised the demon. 



A NEW LAW OF GRAVITATION 119 

Do not imagine that this preliminary change of con- 
ception carries us very far towards an explanation of 
gravitation. We are not seeking an explanation; we 
are seeking a picture. And this picture of world- 
curvature (hard though it may seem) is more graspable 
than an elusive tug which flits from one object to 
another according to the point of view chosen. 

A New Law of Gravitation. Having found a new pic- 
ture of gravitation, we require a new law of gravitation; 
for the Newtonian law told us the arcounr. of the tug 
and there is now no tug to be considered. Since the 
phenomenon is now pictured as curvature the new law 
must say something about curvature. Evidently it must 
be a law governing and limiting the possible curvature 
of space-time. 

There are not many things which can be said about 
curvature — not many of a general character. So that 
when Einstein felt this urgency to say something about 
curvature, he almost automatically said the right thing. 
I mean that there was only one limitation or law that 
suggested itself as reasonable, and that law has proved 
to be right when tested by observation. 

Some of you may feel that you could never bring your 
minds to conceive a curvature of space, let alone of 
space-time; others may feel that, being familiar with 
the bending of a two-dimensional surface, there is no 
insuperable difficulty in imagining something similar for 
three or even four dimensions. I rather think that 
the former have the best of it, for at least they escape 
being misled by their preconceptions. I have spoken of 
a "picture", but it is a picture that has to be described 
analytically rather than conceived vividly. Our ordinary 
conception of curvature is derived from surfaces, i.e. 



120 GRAVITATION— THE LAW 

two-dimensional manifolds embedded in a three-dimen- 
sional space. The absolute curvature at any point is 
measured by a single quantity called the radius of spheri- 
cal curvature. But space-time is a four-dimensional 
manifold embedded in — well, as many dimensions as it 
can find new ways to twist about in. Actually a four- 
dimensional manifold is amazingly ingenious in discover- 
ing new kinds of contortion, and its invention is not 
exhausted until it has been provided with six extra 
dimensions, making ten dimensions in all. Moreover, 
twenty distinct measures are required at each point to 
specify the particular sort and amount of twistiness 
there. These measures are called coefficients of curva- 
ture. Ten of the coefficients stand out more prominently 
than the other ten. 

Einstein's law of gravitation asserts that the ten prin- 
cipal coefficients of curvature are zero in empty space. 

If there were no curvature, i.e. if all the coefficients 
were zero, there would be no gravitation. Bodies would 
move uniformly in straight lines. If curvature were 
unrestricted, i.e. if all the coefficients had unpredictable 
values, gravitation would operate arbitrarily and with- 
out law. Bodies would move just anyhow. Einstein 
takes a condition midway between; ten of the coefficients 
are zero and the other ten are arbitrary. That gives 
a world containing gravitation limited by a law. The 
coefficients are naturally separated into two groups of 
ten, so that there is no difficulty in choosing those which 
are to vanish. 

To the uninitiated it may seem surprising that an 
exact law of Nature should leave some of the coefficients 
arbitrary. But we need to leave something over to be 
settled when we have specified the particulars of the 
problem to which it is proposed to apply the law. A 



A NEW LAW OF GRAVITATION 121 

general law covers an infinite number of special cases. 
The vanishing of the ten principal coefficients occurs 
everywhere in empty space whether there is one gravi- 
tating body or many. The other ten coefficients vary 
according to the special case under discussion. This may 
remind us that after reaching Einstein's law of gravi- 
tation and formulating it mathematically, it is still a very 
long step to reach its application to even the simplest 
practical problem. However, by this time many hun- 
dreds of readers must have gone carefully through the 
mathematics; so we may rest assured that there has 
been no mistake. After this work has been done it 
becomes possible to verify that the law agrees with 
observation. It is found that it agrees with Newton's 
law to a very close approximation so that the main 
evidence for Einstein's law is the same as the evidence 
for Newton's law; but there are three crucial astro- 
nomical phenomena in which the difference is large 
enough to be observed. In these phenomena the obser- 
vations support Einstein's law against Newton's.* 

It is essential to our faith in a theory that its predic- 
tions should accord with observation, unless a reasonable 
explanation of the discrepancy is forthcoming; so that 
it is highly important that Einstein's law should have 
survived these delicate astronomical tests in which New- 
ton's law just failed. But our main reason for reject- 
ing Newton's law is not its imperfect accuracy as shown 
by these tests; it is because it does not contain the 
kind of information about Nature that we want to 
know now that we have an ideal before us which was 
not in Newton's mind at all. We can put it this way. 

* One of the tests — a shift of the spectral lines to the red in the sun 
and stars as compared with terrestrial sources — is a test of Einstein's 
theory rather than of his law. 



122 GRAVITATION— THE LAW 

Astronomical observations show that within certain 
limits of accuracy both Einstein's and Newton's laws 
are true. In confirming (approximately) Newton's law, 
we are confirming a statement as to what the appear- 
ances would be when referred to one particular space- 
time frame. No reason is given for attaching any 
fundamental importance to this frame. In confirming 
(approximately) Einstein's law, we are confirming a 
statement about the absolute properties of the world, 
true for all space-time frames. For those who are try- 
ing to get beneath the appearances Einstein's statement 
necessarily supersedes Newton's; it extracts from the 
observations a result with physical meaning as opposed 
to a mathematical curiosity. That Einstein's law has 
proved itself the better approximation encourages us in 
our opinion that the quest of the absolute is the best way 
to understand the relative appearances; but had the suc- 
cess been less immediate, we could scarcely have turned 
our back on the quest. 

I cannot but think that Newton himself would rejoice 
that after 200 years the "ocean of undiscovered truth" 
has rolled back another stage. I do not think of him as 
censorious because we will not blindly apply his formula 
regardless of the knowledge that has since accumulated 
and in circumstances that he never had the opportunity 
of considering. 

I am not going to describe the three tests here, since 
they are now well known and will be found in any of 
the numerous guides to relativity; but I would refer to 
the action of gravitation on light concerned in one of 
them. Light-waves in passing a massive body such as 
the sun are deflected through a small angle. This is 
additional evidence that the Newtonian picture of 
gravitation as a tug is inadequate. You cannot deflect 



THE LAW OF MOTION 123 

waves by tugging at them, and clearly another repre- 
sentation of the agency which deflects them must be 
found. 

The Law of Motion. I must now ask you to let your 
mind revert to the time of your first introduction to 
mechanics before your natural glimmerings of the truth 
were sedulously uprooted by your teacher. You were 
taught the First Law of Motion — 

"Every body continues in its state of rest or uniform 
motion in a straight line, except in so far as it may be 
compelled to change that state by impressed forces." 

Probably you had previously supposed that motion 
was something which would exhaust itself; a bicycle 
stops of its own accord if you do not impress force to 
keep it going. The teacher rightly pointed out the 
resisting forces which tend to stop the bicycle; and he 
probably quoted the example of a stone skimming over 
ice to show that when these interfering forces are re- 
duced the motion lasts much longer. But even ice offers 
some frictional resistance. Why did not the teacher do 
the thing thoroughly and abolish resisting forces alto- 
gether, as he might easily have done by projecting the 
stone into empty space? Unfortunately in that case 
its motion is not uniform and rectilinear; the stone 
describes a parabola. If you raised that objection you 
would be told that the projectile was compelled to 
change its state of uniform motion by an invisible force 
called gravitation. How do we know that this invisible 
force exists? Why! because if the force did not exist 
the projectile would move uniformly in a straight line. 

The teacher is not playing fair. He is determined to 
have his uniform motion in a straight line, and if we 
point out to him bodies which do not follow his rule 



124 GRAVITATION— THE LAW 

he blandly invents a new force to account for the devia- 
tion. We can improve on his enunciation of the First 
Law of Motion. What he really meant was — 

"Every body continues in its state of rest or uniform 
motion in a straight line, except in so far as it doesn't." 

Material frictions and reactions are visible and abso- 
lute interferences which can change the motion of a 
body. I have nothing to say against them. The mole- 
cular battering can be recognised by anyone who looks 
deeply into the phenomenon no matter what his frame 
of reference. But when there is no such indication of 
disturbance the whole procedure becomes arbitrary. On 
no particular grounds the motion is divided into two 
parts, one of which is attributed to a passive tendency 
of the body called inertia and the other to an interfer- 
ing field of force. The suggestion that the body really 
wanted to go straight but some mysterious agent made 
it go crooked is picturesque but unscientific. It makes 
two properties out of one; and then we wonder why they 
are always proportional to one another — why the gravi- 
tational force on different bodies is proportional to 
their inertia or mass. The dissection becomes untenable 
when we admit that all frames of reference are on the 
same footing. The projectile which describes a parabola 
relative to an observer on the earth's surface describes 
a straight line relative to the man in the lift. Our 
teacher will not easily persuade the man in the lift who 
sees the apple remaining where he released it, that the 
apple really would of its own initiative rush upwards 
were it not that an invisible tug exactly counteracts this 
tendency.* 

Einstein's Law of Motion does not recognise this 
dissection. There are certain curves which can be 

* The reader will verify tkat this is the doctrine the teacher would have 
to inculcate if he went as a missionary to the men in the lift. 



THE LAW OF MOTION 125 

defined on a curved surface without reference to any 
frame or system of partitions, viz. the geodesies or 
shortest routes from one point to another. The geo- 
desies of our curved space-time supply the natural tracks 
which particles pursue if they are undisturbed. 

We observe a planet wandering round the sun in an 
elliptic orbit. A little consideration will show that if we 
add a fourth dimension (time), the continual moving on 
in the time-dimension draws out the ellipse into a helix. 
Why does the planet take this spiral track instead of 
going straight? It is because it is following the shortest 
track; and in the distorted geometry of the curved 
region round the sun the spiral track is shorter than any 
other between the same points. You see the great 
change in our view. The Newtonian scheme says that 
the planet tends to move in a straight line, but the sun's 
gravity pulls it away. Einstein says that the planet tends 
to take the shortest route and does take it. 

That is the general idea, but for the sake of accuracy 
I must make one rather trivial correction. The planet 
takes the longest route. 

You may remember that points along the track of 
any material body (necessarily moving with a speed less 
than the velocity of light) are in the absolute past or 
future of one another; they are not absolutely ''else- 
where". Hence the length of the track in four dimensions 
is made up of time-like relations and must be measured 
in time-units. It is in fact the number of seconds 
recorded by a clock carried on a body which describes 
the track.* This may be different from the time re- 

* It may be objected that you cannot make a clock follow an arbitrary 
curved path without disturbing it by impressed forces (e.g. molecular 
hammering). But this difficulty is precisely analogous to the difficulty 
of measuring the length of a curve with a rectilinear scale, and is sur- 
mounted in the same way. The usual theory of "rectification of curves" 
applies to these time-tracks as well as to space-curves. 



126 GRAVITATION— THE LAW 

corded by a clock which has taken some other route 
between the same terminal points. On p. 39 we con- 
sidered two individuals whose tracks had the same 
terminal points; one of them remained at home on the 
earth and the other travelled at high speed to a distant 
part of the universe and back. The first recorded a 
lapse of 70 years, the second of one year. Notice that 
it is the man who follows the undisturbed track of the 
earth who records or lives the longest time. The man 
whose track was violently dislocated when he reached 
the limit of his journey and started to come back again 
lived only one year. There is no limit to this reduction; 
as the speed of the traveller approaches the speed of 
light the time recorded diminishes to zero. There is no 
unique shortest track; but the longest track is unique. 
If instead of pursuing its actual orbit the earth made a 
wide sweep which required it to travel with the velocity 
of light, the earth could get from 1 January 1927 to 1 
January 1928 in no time, i.e. no time as recorded by an 
observer or clock travelling with it, though it would be 
reckoned as a year according to "Astronomer Royal's 
time". The earth does not do this, because it is a rule 
of the Trade Union of matter that the longest possible 
time must be taken over every job. 

Thus in calculating astronomical orbits and in similar 
problems two laws are involved. We must first cal- 
culate the curved form of space-time by using Einstein's 
law of gravitation, viz. that the ten principal curva- 
tures are zero. We next calculate how the planet moves 
through the curved region by using Einstein's law of 
motion, viz. the law of the longest track. Thus far the 
procedure is analogous to calculations made with New- 
ton's law of gravitation and Newton's law of motion. 
But there is a remarkable addendum which applies only 



THE LAW OF MOTION 127 

to Einstein's laws. Einstein's law of motion can be 
deduced from his law of gravitation. The prediction of 
the track of a planet although divided into two stages 
for convenience rests on a single law. 

I should like to show you in a general way how it is 
possible for a law controlling the curvature of empty 
space to determine the tracks of particles without being 
supplemented by any other conditions. Two "particles" in 
the four-dimensional world are shown in Fig. 5, namely 
yourself and myself. We are not empty space so there is 




— ^-"^ 



Fig. 5 

no limit to the kind of curvature entering into our com- 
position; in fact our unusual sort of curvature is what 
distinguishes us from empty space. We are, so to 
speak, ridges in the four-dimensional world where it is 
gathered into a pucker. The pure mathematician in his 
unflattering language would describe us as "singulari- 
ties". These two non-empty ridges are joined by empty 
space, which must be free from those kinds of curva- 
ture described by the ten principal coefficients. Now 
it is common experience that if we introduce local 
puckers into the material of a garment, the remainder 
has a certain obstinacy and will not lie as smoothly as 



128 GRAVITATION— THE LAW 

we might wish. You will realise the possibility that, 
given two ridges as in Fig. 5, it may be impossible to 
join them by an intervening valley without the illegal 
kind of curvature. That turns out to be the case. Two 
perfectly straight ridges alone in the world cannot be 
properly joined by empty space and therefore they can- 
not occur alone. But if they bend a little towards one 
another the connecting region can lie smoothly and sat- 
isfy the law of curvature. If they bend too much the 
illegal puckering reappears. The law of gravitation is 
a fastidious tailor who will not tolerate wrinkles (except 
of a limited approved type) in the main area of the 
garment; so that the seams are required to take courses 
which will not cause wrinkles. You and I have to sub- 
mit to this and so our tracks curve towards each other. 
An onlooker will make the comment that here is an 
illustration of the law that two massive bodies attract 
each other. 

We thus arrive at another but equivalent conception 
of how the earth's spiral track through the four-dimen- 
sional world is arrived at. It is due to the necessity of 
arranging two ridges (the solar track and the earth's 
track) so as not to involve a wrong kind of curvature in 
the empty part of the world. The sun as the more 
pronounced ridge takes a nearly straight track; but the 
earth as a minor ridge on the declivities of the solar 
ridge has to twist about considerably. 

Suppose the earth were to defy the tailor and take a 
straight track. That would make a horrid wrinkle in the 
garment; and since the wrinkle is inconsistent with the 
laws of empty space, something must be there — where 
the wrinkle runs. This "something" need not be matter 
in the restricted sense. The things which can occupy 
space so that it is not empty in the sense intended in 



RELATIVITY OF ACCELERATION 129 

Einstein's law, are mass (or its equivalent energy) 
momentum and stress (pressure or tension). In this case 
the wrinkle might correspond to stress. That is reason- 
able enough. If left alone the earth must pursue its 
proper curved orbit; but if some kind of stress or pres- 
sure were inserted between the sun and earth, it might 
well take another course. In fact if we were to observe 
one of the planets rushing off in a straight track, New- 
tonians and Einsteinians alike would infer that there 
existed a stress causing this behaviour. It is true that 
causation has apparently been turned topsy-turvy; ac- 
cording to our theory the stress seems to be caused by 
the planet taking the wrong track, whereas we usually 
suppose that the planet takes the wrong track because it 
is acted on by the stress. But that is a harmless accident 
common enough in primary physics. The discrimination 
between cause and effect depends on time's arrow and 
can only be settled by reference to entropy. We need 
not pay much attention to suggestions of causation aris- 
ing in discussions of primary laws which, as likely as 
not, are contemplating the world upside down. 

Although we are here only at the beginning of Ein- 
stein's general theory I must not proceed further into 
this very technical subject. The rest of this chapter will 
be devoted to elucidation of more elementary points. 

Relativity of Acceleration. The argument in this chapter 
rests on the relativity of acceleration. The apple had an 
acceleration of 32 feet per second per second relative to 
the ordinary observer, but zero acceleration relative to 
the man in the lift. We ascribe to it one acceleration or 
the other according to the frame we happen to be using, 
but neither is to be singled out and labelled "true" 
or absolute acceleration. That led us to reject the 



i 3 o GRAVITATION— THE LAW 

Newtonian conception which singled out 32 feet per 
second per second as the true acceleration and invented 
a disturbing agent of this particular degree of strength. 

It will be instructive to consider an objection brought, 
I think, originally by Lenard. A train is passing through 
a station at 60 miles an hour. Since velocity is relative, 
it does not matter whether we say that the train is 
moving at 60 miles an hour past the station or the 
station is moving at 60 miles an hour past the train. 
Now suppose, as sometimes happens in railway acci- 
dents, that this motion is brought to a standstill in a 
few seconds. There has been a change of velocity or 
acceleration — a term which includes deceleration. If 
acceleration is relative this may be described indiffer- 
ently as an acceleration of the train (relative to the sta- 
tion) or an acceleration of the station (relative to the 
train). Why then does it injure the persons in the train 
and not those in the station? 

Much the same point was put to me by one of my 
audience. "You must find the journey between Cam- 
bridge and Edinburgh very tiring. I can understand 
the fatigue, if you travel to Edinburgh; but why should 
you get tired if Edinburgh comes to you?" The answer 
is that the fatigue arises from being shut up in a box 
and jolted about for nine hours; and it makes no differ- 
ence whether in the meantime I move to Edinburgh or 
Edinburgh moves to me. Motion does not tire anybody. 
With the earth as our vehicle we are travelling at 20 
miles a second round the sun; the sun carries us at 12 
miles a second through the galactic system; the galactic 
system bears us at 250 miles a second amid the spiral 
nebulae; the spiral nebulae. ... If motion could tire, 
we ought to be dead tired. 

Similarly change of motion or acceleration does not 



RELATIVITY OF ACCELERATION 131 

injure anyone, even when it is (according to the New- 
tonian view) an absolute acceleration. We do not even 
feel the change of motion as our earth takes the curve 
round the sun. We feel something when a railway train 
takes a curve, but what we feel is not the change of 
motion nor anything which invariably accompanies 
change of motion; it is something incidental to the 
curved track of the train but not to the curved track of 
the earth. The cause of injury in the railway accident 
is easily traced. Something hit the train; that is to say, 
the train was bombarded by a swarm of molecules and 
the bombardment spread all the way along it. The 
cause is evident — gross, material, absolute — recognised 
by everyone, no matter what his frame of reference, 
as occurring in the train not the station. Besides injur- 
ing the passengers this cause also produced the relative 
acceleration of the train and station — an effect which 
might equally well have been produced by molecular 
bombardment of the station, though in this case it was 

not 

The critical reader will probably pursue his objection. 
"Are you not being paradoxical when you say that a 
molecular bombardment of the train can cause an accel- 
eration of the station — and in fact of the earth and the 
rest of the universe? To put it mildly, relative accelera- 
tion is a relation with two ends to it, and we may at 
first seem to have an option which end we shall grasp 
it by; but in this case the causation (molecular bom- 
bardment) clearly indicates the right end to take hold 
of, and you are merely spinning paradoxes when you 
insist on your liberty to take hold of the other." 

If there is an absurdity in taking hold of the wrong 
end of the relation it has passed into our current 
speech and thought. Your suggestion is in fact more 



132 GRAVITATION— THE LAW 

revolutionary than anything Einstein has ventured to 
advocate. Let us take the problem of a falling stone. 
There is a relative acceleration of 32 feet per second 
per second — of the stone relative to ourselves or of our- 
selves relative to the stone. Which end of the relation 
must we choose? The one indicated by molecular bom- 
bardment? Well, the stone is not bombarded; it is 
falling freely in vacuo. But we are bombarded by the 
molecules of the ground on which we stand. Therefore 
it is we who have the acceleration; the stone has zero 
acceleration, as the man in the lift supposed. Your sug- 
gestion makes out the frame of the man in the lift to 
be the only legitimate one; I only went so far as to 
admit it to an equality with our own customary frame. 

Your suggestion would accept the testimony of the 
drunken man who explained that u the paving-stone got 
up and hit him" and dismiss the policeman's account of 
the incident as "merely spinning paradoxes". What 
really happened was that the paving-stone had been 
pursuing the man through space with ever-increasing 
velocity, shoving the man in front of it so that they kept 
the same relative position. Then, through an unfor- 
tunate wobble of the axis of the man's body, he failed 
to increase his speed sufficiently, with the result that 
the paving-stone overtook him and came in contact with 
his head. That, please understand, is your suggestion; 
or rather the suggestion which I have taken the liberty 
of fathering on you because it is the outcome of a very 
common feeling of objection to the relativity theory. 
Einstein's position is that whilst this is a perfectly 
legitimate way of looking at the incident the more usual 
account given by the policeman is also legitimate; and 
he endeavours like a good magistrate to reconcile them 
both. 



TIME GEOMETRY 133 

Time Geometry. Einstein's law of gravitation controls 
a geometrical quantity curvature in contrast to Newton's 
law which controls a mechanical quantity force. To 
understand the origin of this geometrisation of the world 
in the relativity theory we must go back a little. 

The science which deals with the properties of space 
is called geometry. Hitherto geometry has not included 
time in its scope. But now space and time are so inter- 
locked that there must be one science — a somewhat 
extended geometry — embracing them both. Three- 
dimensional space is only a section cut through four- 
dimensional space-time, and moreover sections cut in 
different directions form the spaces of different 
observers. We can scarcely maintain that the study of 
a section cut in one special direction is the proper sub- 
ject-matter of geometry and that the study of slightly 
different sections belongs to an altogether different 
science. Hence the geometry of the world is now con- 
sidered to include time as well as space. Let us follow 
up the geometry of time. 

You will remember that although space and time are 
mixed up there is an absolute distinction between a 
spatial and a temporal relation of two events. Three 
events will form a space-triangle if the three sides 
correspond to spatial relations — if the three events are 
absolutely elsewhere with respect to one another.* 
Three events will form a time-triangle if the three sides 
correspond to temporal relations — if the three events 
are absolutely before or after one another. (It is pos- 
sible also to have mixed triangles with two sides time-like 
and one space-like, or vice versa.) A well-known law 
of the space-triangle is that any two sides are together 

* This would be an instantaneous space-triangle. An enduring triangle 
is a kind of four-dimensional prism. 



134 GRAVITATION— THE LAW 

greater than the third side. There is an analogous, but 
significantly different, law for the time-triangle, viz. two 
of the sides (not any two sides) are together less than 
the third side. It is difficult to picture such a triangle 
but that is the actual fact. 

Let us be quite sure that we grasp the precise mean- 
ing of these geometrical propositions. Take first the 
space-triangle. The proposition refers to the lengths of 
the sides, and it is well to recall my imaginary discus- 
sion with two students as to how lengths are to be 
measured (p. 23). Happily there is no ambiguity 
now, because the triangle of three events determines a 
plane section of the world, and it is only for that mode 
of section that the triangle is purely spatial. The propo- 
sition then expresses that 

"If you measure with a scale from A to B and from 
B to C the sum of your readings will be greater than the 
reading obtained by measuring with a scale from A to C." 

For a time-triangle the measurements must be made 
with an instrument which can measure time, and the 
proposition then expresses that 

"If you measure with a clock from A to B and from 
B to C the sum of your readings will be less than the 
reading obtained by measuring with a clock from A to C." 

In order to measure from an event A to an event B 
with a clock you must make an adjustment of the clock 
analogous to orienting a scale along the line AB. What 
is this analogous adjustment? The purpose in either 
case is to bring both A and B into the immediate 
neighbourhood of the scale or clock. For the clock that 
means that after experiencing the event A it must travel 
with the appropriate velocity needed to reach the locality 
of B just at the moment that B happens. Thus the 
velocity of the clock is prescribed. One further point 



TIME GEOMETRY 135 

should be noticed. After measuring with a scale from 
A to B you can turn your scale round and measure from 
B to A, obtaining the same result. But you cannot turn 
a clock round, i.e. make it go backwards in time. That 
is important because it decides which two sides are less 
than the third side. If you choose the wrong pair the 
enunciation of the time proposition refers to an im- 
possible kind of measurement and becomes meaningless. 

You remember the traveller (p. 39) who went off 
to a distant star and returned absurdly young. He was 
a clock measuring two sides of a time-triangle. He 
recorded less time than the stay-at-home observer who 
was a clock measuring the third side. Need I defend 
my calling him a clock? We are all of us clocks whose 
faces tell the passing years. This comparison was simply 
an example of the geometrical proposition about time- 
triangles (which in turn is a particular case of Einstein's 
law of longest track). The result is quite explicable in 
the ordinary mechanical way. All the particles in the 
traveller's body increase in mass on account of his high 
velocity according to the law already discussed and 
verified by experiment. This renders them more slug- 
gish, and the traveller lives more slowly according to 
terrestrial time-reckoning. However, the fact that the 
result is reasonable and explicable does not render it the 
less true as a proposition of time geometry. 

Our extension of geometry to include time as well as 
space will not be a simple addition of an extra dimension 
to Euclidean geometry, because the time propositions, 
though analogous, are not identical with those which 
Euclid has given us for space alone. Actually the dif- 
ference between time geometry and space geometry is 
not very profound, and the mathematician easily glides 
over it by a discrete use of the symbol V-i. We still 



i 3 6 GRAVITATION— THE LAW 

call (rather loosely) the extended geometry Euclidean; 
or, if it is necessary to emphasise the distinction, we 
call it hyperbolic geometry. The term non-Euclidean 
geometry refers to a more profound change, viz. that 
involved in the curvature of space and time by which 
we now represent the phenomenon of gravitation. We 
start with Euclidean geometry of space, and modify it 
in a comparatively simple manner when the time-dimen- 
sion is added; but that still leaves gravitation to be 
reckoned with, and wherever gravitational effects are 
observable it is an indication that the extended Euclidean 
geometry is not quite exact, and the true geometry is a 
non-Euclidean one — appropriate to a curved region as 
Euclidean geometry is to a flat region. 

Geometry and Mechanics. The point that deserves special 
attention is that the proposition about time-triangles is 
a statement as to the behaviour of clocks moving with 
different velocities. We have usually regarded the 
behaviour of clocks as coming under the science of 
mechanics. We found that it was impossible to confine 
geometry to space alone, and we had to let it expand a 
little. It has expanded with a vengeance and taken a 
big slice out of mechanics. There is no stopping it, and 
bit by bit geometry has now swallowed up the whole of 
mechanics. It has also made some tentative nibbles at 
electromagnetism. An ideal shines in front of us, far 
ahead perhaps but irresistible, that the whole of our 
knowledge of the physical world may be unified into a 
single science which will perhaps be expressed in terms 
of geometrical or quasi-geometrical conceptions. Why 
not? All the knowledge is derived from measurements 
made with various instruments. The instruments used 
in the different fields of inquiry are not fundamentally 



GEOMETRY AND MECHANICS 137 

unlike. There is no reason to regard the partitions of 
the sciences made in the early stages of human thought 
as irremovable. 

But mechanics in becoming geometry remains none 
the less mechanics. The partition between mechanics 
and geometry has broken down and the nature of each 
of them has diffused through the whole. The apparent 
supremacy of geometry is really due to the fact that it 
possesses the richer and more adaptable vocabulary; 
and since after the amalgamation we do not need the 
double vocabulary the terms employed are generally 
taken from geometry. But besides the geometrisation of 
mechanics there has been a mechanisation of geometry. 
The proposition about the space-triangle quoted above 
was seen to have grossly material implications about the 
behaviour of scales which would not be realised by any- 
one who thinks of it as if it were a proposition of pure 
mathematics. 

We must rid our minds of the idea that the word 
space in science has anything to do with void. As pre- 
viously explained it has the other meaning of distance, 
volume, etc., quantities expressing physical measure- 
ment just as much as force is a quantity expressing 
physical measurement. Thus the (rather crude) state- 
ment that Einstein's theory reduces gravitational force 
to a property of space ought not to arouse misgiving. 
In any case the physicist does not conceive of space 
as void. Where it is empty of all else there is still the 
aether. Those who for some reason dislike the word 
aether, scatter mathematical symbols freely through the 
vacuum, and I presume that they must conceive some 
kind of characteristic background for these symbols. I 
do not think any one proposes to build even so relative 
and elusive a thing as force out of entire nothingness. 



Chapter VII 

GRAVITATION— THE EXPLANATION 

The Law of Curvature. Gravitation can be explained. 
Einstein's theory is not primarily an explanation of 
gravitation. When he tells us that the gravitational field 
corresponds to a curvature of space and time he is giv- 
ing us a picture. Through a picture we gain the insight 
necessary to deduce the various observable consequences. 
There remains, however, a further question whether 
any reason can be given why the state of things pictured 
should exist. It is this further inquiry which is meant 
when we speak of "explaining" gravitation in any far- 
reaching sense. 

At first sight the new picture does not leave very 
much to explain. It shows us an undulating hum- 
mocky world, whereas a gravitationless world would be 
plane and uniform. But surely a level lawn stands more 
in need of explanation than an undulating field, and a 
gravitationless world would be more difficult to account 
for than a world with gravitation. We are hardly called 
upon to account for a phenomenon which could only 
be absent if (in the building of the world) express pre- 
cautions were taken to exclude it. If the curvature were 
entirely arbitrary this would be the end of the explana- 
tion; but there is a law of curvature — Einstein's law of 
gravitation — and on this law our further inquiry must 
be focussed. Explanation is needed for regularity, not 
for diversity; and our curiosity is roused, not by the 
diverse values of the ten subsidiary coefficients of curva- 
ture which differentiate the world from a flat world, 
but by the vanishing everywhere of the ten principal 
coefficients. 

138 



THE LAW OF CURVATURE 139 

All explanations of gravitation on Newtonian lines 
have endeavoured to show why something (which I have 
disrespectfully called a demon) is present in the world. 
An explanation on the lines of Einstein's theory must 
show why something (which we call principal curvature) 
is excluded from the world. 

In the last chapter the law of gravitation was stated 
in the form — the ten principal coefficients of curvature 
vanish in empty space. I shall now restate it in a slightly 
altered form — 

The radius of spherical* curvature of every three-di- 
mensional section of the world, cut in any direction at any 
point of empty space, is always the same constant length. 

Besides the alteration of form there is actually a little 
difference of substance between the two enunciations; 
the second corresponds to a later and, it is believed, more 
accurate formula given by Einstein a year or two after 
his first theory. The modification is. made necessary by 
our realisation that space is finite but unbounded (p. 
80). The second enunciation would be exactly equiva- 
lent to the first if for "same constant length" we read 
"infinite length". Apart from very speculative esti- 
mates we do not know the constant length referred to, 
but it must certainly be greater than the distance of the 
furthest nebula, say io 20 miles. A distinction between 
so great a length and infinite length is unnecessary in 
most of our arguments and investigations, but it is 
necessary in the present chapter. 

♦Cylindrical curvature of the world has nothing to do with gravita- 
tion, nor so far as we know with any other phenomenon. Anything 
drawn on the surface of a cylinder can be unrolled into a flat map without 
distortion, but the curvature introduced in the last chapter was intended 
to account for the distortion which appears in our customary flat map; it 
is therefore curvature of the type exemplified by a sphere, not a cylinder. 



140 GRAVITATION— THE EXPLANATION 

We must try to reach the vivid significance which 
lies behind the obscure phraseology of the law. Suppose 
that you are ordering a concave mirror for a telescope. 
In order to obtain what you want you will have to 
specify two lengths (i) the aperture, and (2) the radius 
of curvature. These lengths both belong to the mirror — 
both are necessary to describe the kind of mirror you 
want to purchase — but they belong to it in different 
ways. You may order a mirror of 100 foot radius of 
curvature and yet receive it by parcel post. In a certain 
sense the 100 foot length travels with the mirror, but 
it does so in a way outside the cognizance of the postal 
authorities. The 100 foot length belongs especially to 
the surface of the mirror, a two-dimensional continuum; 
space-time is a four-dimensional continuum, and you will 
see from this analogy that there can be lengths belonging 
in this way to a chunk of space-time — lengths having 
nothing to do with the largeness or smallness of the 
chunk, but none the less part of the specification of the 
particular sample. Owing to the two extra dimensions 
there are many more such lengths associated with space- 
time than with the mirror surface. In particular, there 
is not only one general radius of spherical curvature, but 
a radius corresponding to any direction you like to take. 
For brevity I will call this the "directed radius" of the 
world. Suppose now that you order a chunk of space- 
time with a directed radius of 500 trillion miles in one 
direction and 800 trillion miles in another. Nature 
replies "No. We do not stock that. We keep a wide 
range of choice as regards other details of specification; 
but as regards directed radius we have nothing different 
in different directions, and in fact all our goods have the 
one standard radius, x trillion miles." I cannot tell you 
what number to put for x because that is still a secret 
of the firm. 



RELATIVITY OF LENGTH 141 

The fact that this directed radius which, one would 
think, might so easily differ from point to point and 
from direction to direction, has only one standard value 
in the world is Einstein's law of gravitation. From it 
we can by rigorous mathematical deduction work out the 
motions of planets and predict, for example, the eclipses 
of the next thousand years; for, as already explained, 
the law of gravitation includes also the law of motion. 
Newton's law of gravitation is an approximate adapta- 
tion of it for practical calculation. Building up from 
the law all is clear; but what lies beneath it? Why is 
there this unexpected standardisation? That is what we 
must now inquire into. 

Relativity of Length. There is no such thing as absolute 
length; we can only express the length of one thing in 
terms of the length of something else.* And so when 
we speak of the length of the directed radius we mean 
its length compared with the standard metre scale. 
Moreover, to make this comparison, the two lengths 
must lie alongside. Comparison at a distance is as un- 
thinkable as action at a distance; more so, because com- 
parison is a less vague conception than action. We must 
either convey the standard metre to the site of the 
length we are measuring, or we must use some device 
which, we are satisfied, will give the same result as if we 
actually moved the metre rod. 

Now if we transfer the metre rod to another point of 
space and time, does it necessarily remain a metre long? 
Yes, of course it does; so long as it is the standard of 
length it cannot be anything else but a metre. But does 
it really remain the metre that it was? I do not know 

* This relativity with respect to a standard unit is, of course, addi- 
tional to and independent of the relativity with respect to the observer's 
motion treated in chapter n. 



1 42 GRAVITATION— THE EXPLANATION 

what you mean by the question; there is nothing by 
reference to which we could expose delinquencies of the 
standard rod, nothing by reference to which we could 
conceive the nature of the supposed delinquencies. Still 
the standard rod was chosen with considerable care; its 
material was selected to fulfil certain conditions — to be 
affected as little as possible by casual influences such 
as temperature, strain or corrosion, in order that its 
extension might depend only on the most essential char- 
acteristics of its surroundings, present and past.* We 
cannot say that it was chosen to keep the same absolute 
length since there is no such thing known; but it was 

* In so far as these casual influences are not entirely eliminated by 
the selection of material and the precautions in using the rod, appropriate 
corrections must be applied. But the rod must not be corrected for 
essential characteristics of the space it is measuring. We correct the 
reading of a voltmeter for temperature, but it would be nonsensical to 
correct it for effects of the applied voltage. The distinction between 
casual and essential influences — those to be eliminated and those to be 
left in — depends on the intention of the measurements. The measuring 
rod is intended for surveying space, and the essential characteristic of 
space is "metric". It would be absurd to correct the readings of our 
scale to the values they would have had if the space had some other 
metric. The region of the world to which the metric refers may also 
contain an electric field; this will be regarded as a casual characteristic 
since the measuring rod is not intended for surveying electric fields. 
I do not mean that from a broader standpoint the electric field is any less 
essential to the region than its peculiar metric. It would be hard to say 
in what sense it would remain the same region if any of its qualities were 
other than they actually are. This point does not trouble us here, because 
there are vast regions of the world practically empty of all characteristics 
except metric, and it is to these that the law of gravitation is applied both 
in theory and in practice. It has seemed, however, desirable to dwell on 
this distinction between essential and casual characteristics because there 
are some who, knowing that we cannot avoid in all circumstances cor- 
rections for casual influences, regard that as license to adopt any arbi- 
trary system of corrections — a procedure which would merely have the 
effect of concealing what the measures can teach us about essential 
characteristics. 



RELATIVITY OF LENGTH 143 

chosen so that it might not be prevented by casual in- 
fluences from keeping the same relative length — relative 
to what? Relative to some length inalienably associated 
with the region in which it is placed. I can conceive 
of no other answer. An example of such a length 
inalienably associated with a region is the directed radius. 

The long and short of it is that when the standard 
metre takes up a new position or direction it measures 
itself against the directed radius of the world in that 
region and direction, and takes up an extension which 
is a definite fraction of the directed radius. I do not 
see what else it could do. We picture the rod a little 
bewildered in its new surroundings wondering how 
large it ought to be — how much of the unfamiliar terri- 
tory its boundaries ought to take in. It wants to do 
just what it did before. Recollections of the chunk of 
space that it formerly filled do not help, because there 
is nothing of the nature of a landmark. The one thing 
it can recognise is a directed length belonging to the 
region where it finds itself; so it makes itself the same 
fraction of this directed length as it did before. 

If the standard metre is always the same fraction of 
the directed radius, the directed radius is always the 
same number of metres. Accordingly the directed 
radius is made out to have the same length for all 
positions and directions. Hence we have the law of 
gravitation. 

When we felt surprise at finding as a law of Nature 
that the directed radius of curvature was the same for 
all positions and directions, we did not realise that our 
unit of length had already made itself a constant fraction 
of the directed radius. The whole thing is a vicious 
circle. The law of gravitation is — a put-up job. 



144 GRAVITATION— THE EXPLANATION 

This explanation introduces no new hypothesis. In 
saying that a material system of standard specification 
always occupies a constant fraction of the directed radius 
of the region where it is, we are simply reiterating 
Einstein's law of gravitation — stating it in the inverse 
form. Leaving aside for the moment the question 
whether this behaviour of the rod is to be expected or 
not, the law of gravitation assures us that that is the 
behaviour. To see the force of the explanation we 
must, however, realise the relativity of extension. Exten- 
sion which is not relative to something in the surround- 
ings has no meaning. Imagine yourself alone in the 
midst of nothingness, and then try to tell me how large 
you are. The definiteness of extension of the standard 
rod can only be a definiteness of its ratio to some other 
extension. But we are speaking now of the extension 
of a rod placed in empty space, so that every standard 
of reference has been removed except extensions be- 
longing to and implied by the metric of the region. It 
follows that one such extension must appear from our 
measurements to be constant everywhere (homogeneous 
and isotropic) on account of its constant relation to what 
we have accepted as the unit of length. 

We approached the problem from the point of view 
that the actual world with its ten vanishing coefficients 
of curvature (or its isotropic directed curvature) has a 
specialisation which requires explanation; we were then 
comparing it in our minds with a world suggested by 
the pure mathematician which has entirely arbitrary 
curvature. But the fact is that a world of arbitrary 
curvature is a sheer impossibility. If not the directed 
radius, then some other directed length derivable from 
the metric, is bound to be homogeneous and isotropic. 
In applying the ideas of the pure mathematician we 



RELATIVITY OF LENGTH 145 

overlooked the fact that he was imagining a world 
surveyed from outside with standards foreign to it 
whereas we have to do with a world surveyed from 
within with standards conformable to it. 

The explanation of the law of gravitation thus lies in 
the fact that we are dealing with a world surveyed from 
within. From this broader standpoint the foregoing 
argument can be generalised so that it applies not only 
to a survey with metre rods but to a survey by optical 
methods, which in practice are generally substituted as 
equivalent. When we recollect that surveying apparatus 
can have no extension in itself but only in relation to the 
world, so that a survey of space is virtually a self-com- 
parison of space, it is perhaps surprising that such a 
self-comparison should be able to show up any hetero- 
geneity at all. It can in fact be proved that the metric 
of a two-dimensional or a three-dimensional world sur- 
veyed from within is necessarily uniform. With four or 
more dimensions heterogeneity becomes possible, but it 
is a heterogeneity limited by a law which imposes some 
measure of homogeneity. 

I believe that this has a close bearing on the rather 
heterodox views of Dr. Whitehead on relativity. He 
breaks away from Einstein because he will not admit 
the non-uniformity of space-time involved in Einstein's 
theory. "I deduce that our experience requires and 
exhibits a basis of uniformity, and that in the case of 
nature this basis exhibits itself as the uniformity of 
spatio-temporal relations. This conclusion entirely cuts 
away the casual heterogeneity of these relations which 
is the essential of Einstein's later theory."* But we now 
see that Einstein's theory asserts a casual heterogeneity 

*A. N. Whitehead, The Principle of Relativity, Preface. 



146 GRAVITATION—THE EXPLANATION 

of only one set of ten coefficients and complete uniform- 
ity of the other ten. It therefore does not leave us with- 
out the basis of uniformity of which Whitehead in his 
own way perceived the necessity. Moreover, this uni- 
formity is not the result of a law casually imposed on the 
world; it is inseparable from the conception of survey of 
the world from within — which is, I think, just the con- 
dition that Whitehead would demand. If the world of 
space-time had been of two or of three dimensions 
Whitehead would have been entirely right; but then 
there could have been no Einstein theory of gravitation 
for him to criticise. Space-time being four-dimensional, 
we must conclude that Whitehead discovered an im- 
portant truth about uniformity but misapplied it. 

The conclusion that the extension of an object in any 
direction in the four-dimensional world is determined by 
comparison with the radius of curvature in that direction 
has one curious consequence. So long as the direction 
in the four-dimensional world is space-like, no difficulty 
arises. But when we pass over to time-like directions 
(within the cone of absolute past or future) the directed 
radius is an imaginary length. Unless the object 
ignores the warning symbol V — i it has no standard 
of reference for settling its time extension. It has no 
standard duration. An electron decides how large it 
ought to be by measuring itself against the radius of 
the world in its space-directions. It cannot decide how 
long it ought to exist because there is no real radius of 
the world in its time-direction. Therefore it just goes on 
existing indefinitely. This is not intended to be a rigor- 
ous proof of the immortality of the electron — subject 
always to the condition imposed throughout these 
arguments that no agency other than metric interferes 
with the extension. But it shows that the electron 



PREDICTIONS FROM THE LAW 147 

behaves in the simple way which we might at least hope 
to find.* 

Predictions from the Law. I suppose that it is at first 
rather staggering to find a law supposed to control the 
movements of stars and planets turned into a law 
finicking with the behaviour of measuring rods. But 
there is no prediction made by the law of gravitation in 
which the behaviour of measuring appliances does not 
play an essential part. A typical prediction from the law 
is that pn a certain date 384,400,000 metre rods laid 
end to end would stretch from the earth to the moon. 
We may use more circumlocutory language, but that is 
what is meant. The fact that in testing the prediction 
we shall trust to indirect evidence, not carrying out the 
whole operation literally, is not relevant; the prophecy 
is made in good faith and not with the intention of tak- 
ing advantage of our remissness in checking it. 

We have condemned the law of gravitation as a put- 
up job. You will want to know how after such a dis- 
creditable exposure it can still claim to predict eclipses 
and other events which come off. 

A famous philosopher has said — 

"The stars are not pulled this way and that by 
mechanical forces; theirs is a free motion. They go on 
their way, as the ancients said, like the blessed gods." f 

This sounds particularly foolish even for a philo- 
sopher; but I believe that there is a sense in which it is 
true. 

* On the other hand a quantum (see chapter ix) has a definite 
periodicity associated with it, so that it must be able to measure itself 
against a time-extension. Anyone who contemplates the mathematical 
equations of the new quantum theory will see abundant evidence of the 
battle with the intervening symbol V — *• 

t Hegel, Werke (1842 Ed.), Bd. 7, Abt. 1, p. 97. 



i 4 8 GRAVITATION— THE EXPLANATION 

We have already had three versions of what the earth 
is trying to do when it describes its elliptic orbit around 
the sun. 

(i) It is trying to go in a straight line but it is 
roughly pulled away by a tug emanating from the sun. 

(2) It is taking the longest possible route through 
the curved space-time around the sun. 

(3) It is accommodating its track so as to avoid 
causing any illegal kind of curvature in the empty space 
around it. 

We now add a fourth version. 

(4) The earth goes anyhow it likes. 

It is not a long step from the third version to the 
fourth now that we have seen that the mathematical 
picture of empty space containing "illegal" curvature 
is a sheer impossibility in a world surveyed from within. 
For if illegal curvature is a sheer impossibility the earth 
will not have to take any special precautions to avoid 
causing it, and can do anything it likes. And yet the 
non-occurrence of this impossible curvature is the law 
(of gravitation) by which we calculate the track of the 
earth! 

The key to the paradox is that we ourselves, our 
conventions, the kind of thing that attracts our interest, 
are much more concerned than we realise in any account 
we give of how the objects of the physical world are 
behaving. And so an object which, viewed through our 
frame of conventions, anay seem to be behaving in a 
very special and remarkable way may, viewed according 
to another set of conventions, be doing nothing to excite 
particular comment. This will be clearer if we consider 
a practical illustration, and at the same time defend 
version (4). 



PREDICTIONS FROM THE LAW 149 

You will say that the earth must certainly get into 
the right position for the eclipse next June (1927); so 
it cannot be free to go anywhere it pleases. I can put 
that right. I hold to it that the earth goes anywhere it 
pleases. The next thing is that we must find out where 
it has been pleased to go. The important question for us 
is not where the earth has got to in the inscrutable 
absolute behind the phenomena, but where we shall 
locate it in our conventional background of space and 
time. We must take measurements of its position, for 




Fig. 6 

example, measurements of its distance from the sun. 
In Fig. 6, SSx shows the ridge in the world which we 
recognise as the sun; I have drawn the earth's ridge in 
duplicate (EE 1} EE 2 ) because I imagine it as still un- 
decided which track it will take. If it takes EE ± we lay 
our measuring rods end to end down the ridges and 
across the valley from S ± to E ± , count up the number, 
and report the result as the earth's distance from the 
sun. The measuring rods, you will remember, adjust 
their lengths proportionately to the radius of curvature 
of the world. The curvature along this contour is rather 



150 GRAVITATION— THE EXPLANATION 

large and the radius of curvature small. The rods 
therefore are small, and there will be more of them in 
$i£i than the picture would lead you to expect. If the 
earth chooses to go to E 2 the curvature is less sharp; 
the greater radius of curvature implies greater length 
of the rods. The number needed to stretch from S ± to 
E 2 will not be so great as the diagram at first suggests; 
it will not be increased in anything like the proportion 
of S ± E 2 to S ± E X in the figure. We should not be sur- 
prised if the number turned out to be the same in both 
cases. If so, the surveyor will report the same distance 
of the earth from the sun whether the track is EE ± or 
EE 2 . And the Superintendent of the Nautical Almanac 
who published this same distance some years in advance 
will claim that he correctly predicted where the earth 
would go. 

And so you see that the earth can play truant to any 
extent but our measurements will still report it in the 
place assigned to it by the Nautical Almanac. The 
predictions of that authority pay no attention to the 
vagaries of the god-like earth; they are based on what 
will happen when we come to measure up the path that 
it has chosen. We shall measure it with rods that adjust 
themselves to the curvature of the world. The mathe- 
matical expression of this fact is the law of gravitation 
used in the predictions. 

Perhaps you will object that astronomers do not in 
practice lay measuring rods end to end through inter- 
planetary space in order to find out where the planets 
are. Actually the position is deduced from the light 
rays. But the light as it proceeds has to find out what 
course to take in order to go "straight", in much the 
same way as the metre rod has to find out how far to 
extend. The metric or curvature is a sign-post for the 



PREDICTIONS FROM THE LAW 151 

light as it is a gauge for the rod. The light track is in 
fact controlled by the curvature in such a way that it is 
incapable of exposing the sham law of curvature. And 
so wherever the sun, moon and earth may have got to, 
the light will not give them away. If the law of curva- 
ture predicts an eclipse the light will take such a track 
that there is an eclipse. The law of gravitation is not a 
stern ruler controlling the heavenly bodies; it is a kind- 
hearted accomplice who covers up their delinquencies. 

I do not recommend you to try to verify from Fig. 6 
that the number of rods in SiE t (full line) and SJL 2 
(dotted line) is the same. There are two dimensions of 
space-time omitted in the picture besides the extra dimen- 
sions in which space-time must be supposed to be bent; 
moreover it is the spherical, not the cylindrical, curvature 
which is ,the gauge for the length. It might be an 
instructive, though very laborious, task to make this 
direct verification, but we know beforehand that the 
measured distance of the earth from the sun must be 
the same for either track. The law of gravitation, ex- 
pressed mathematically by G^ u — Xg^ t means nothing 
more nor less than that the unit of length everywhere 
is a constant fraction of the directed radius of the world 
at that point. And as the astronomer who predicts the 
future position of the earth does not assume anything 
more about what the earth will choose to do than is 
expressed in the law G tLV ^=Xg lxl/i so we shall find the 
same position of the earth, if we assume nothing more 
than that the practical unit of length involved in measure- 
ments of the position is a constant fraction of the directed 
radius. We do not need to decide whether the track is 
to be represented by EE ± or EE 2 , and it would convey 
no information as to any observable phenomena if we 
knew the representation. 



152 GRAVITATION— THE EXPLANATION 

I shall have to emphasise elsewhere that the whole of 
our physical knowledge is based on measures and that 
the physical world consists, so to speak, of measure- 
groups resting on a shadowy background that lies 
outside the scope of physics. Therefore in conceiving 
a world which had existence apart from the measure- 
ments that we make of it, I was trespassing outside the 
limits of what we call physical reality. I would not 
dissent from the view that a vagary which by its very 
nature could not be measurable has no claim to a physical 
existence. No one knows what is meant by such a 
vagary. I said that the earth might go anywhere it 
chose, but did not provide a "where" for it to choose; 
since our conception of "where" is based on space 
measurements which were at that stage excluded. But 
I do not think I have been illogical. I am urging that, 
do what it will, the earth cannot get out of the track 
laid down for it by the law of gravitation. In order to 
show this I must suppose that the earth has made the 
attempt and stolen nearer to the sun; then I show that 
our measures conspire quietly to locate it back in its 
proper orbit. I have to admit in the end that the earth 
never was out of its proper orbit;* I do not mind that, 
because meanwhile I have proved my point. The fact 
that a predictable path through space and time is laid 
down for the earth is not a genuine restriction on its 
conduct, but is imposed by the formal scheme in which 
we draw up our account of its conduct. 

* Because I can attach no meaning to an orbit other than an orbit in 
space and time, i.e. as located by measures. But I could not assume that 
the alternative orbit would be meaningless (inconsistent with possible 
measures) until I tried it. 



NON-EMPTY SPACE 153 

Non-Empty Space. The law that the directed radius is 
constant does not apply to space which is not completely 
empty. There is no longer any reason to expect it to 
hold. The statement that the region is not empty means 
that it has other characteristics besides metric, and the 
metre rod can then find other lengths besides curvatures 
to measure itself against. Referring to the earlier (suf- 
ficiently approximate) expression of the law, the ten 
principal coefficients of curvature are zero in empty 
space but have non-zero values in non-empty space. It 
is therefore natural to use these coefficients as a measure 
of the fullness of space. 

One of the coefficients corresponds to mass (or 
energy) and in most practical cases it outweighs the 
others in importance. The old definition of mass as 
"quantity of matter" associates it with a fullness of 
space. Three other coefficients make up the momentum 
— a directed quantity with three independent com- 
ponents. The remaining six coefficients of principal 
curvature make up the stress or pressure-system. Mass, 
momentum and stress accordingly represent the non- 
emptiness of a region in so far as it is able to disturb 
the usual surveying apparatus with which we explore 
space — clocks, scales, light-rays, etc. It should be 
added, however, that this is a summary description and 
not a full account of the non-emptiness, because we 
have other exploring apparatus — magnets, electroscopes, 
etc. — which provide further details. It is usually con- 
sidered that when we use these we are exploring not 
space, but a field in space. The distinction thus created 
is a rather artificial one which is unlikely to be accepted 
permanently. It would seem that the results of ex- 
ploring the world with a measuring scale and a magnetic 
compass respectively ought to be welded together into 



154 GRAVITATION— THE EXPLANATION 

a unified description, just as we have welded together 
results of exploration with a scale and a clock. Some 
progress has been made towards this unification. There 
is, however, a real reason for admitting a partially 
separate treatment; the one mode of exploration deter- 
mines the symmetrical properties and the other the 
antisymmetrical properties of the underlying world- 
structure.* 

Objection has often been taken, especially by philo- 
sophical writers, to the crudeness of Einstein's initial 
requisitions, viz. a clock and a measuring scale. But the 
body of experimental knowledge of the world which 
Einstein's theory seeks to set in order has not come into 
our minds as a heaven-sent inspiration; it is the result 
of a survey in which the clock and the scale have actually 
played the leading part. They may seem very gross 
instruments to those accustomed to the conceptions of 
atoms and electrons, but it is correspondingly gross 
knowledge that we have been discussing in the chapters 
concerned with Einstein's theory. As the relativity 
theory develops, it is generally found desirable to replace 
the clock and scale by the moving particle and light- 
ray as the primary surveying appliances; these are test 
bodies of simpler structure. But they are still gross 
compared with atomic phenomena. The light-ray, for 
instance, is not applicable to measurements so refined 
that the diffraction of light must be taken into account. 
Our knowledge of the external world cannot be divorced 
from the nature of the appliances with which we have 
obtained the knowledge. The truth of the law of gravi- 
tation cannot be regarded as subsisting apart from the 
experimental procedure by which we have ascertained 
its truth. 

* See p. 236. 



NON-EMPTY SPACE 155 

The conception of frames of space and time, and of the 
non-emptiness of the world described as energy, momen- 
tum, etc., is bound up with the survey by gross ap- 
pliances. When they can no longer be supported by 
such a survey, the conceptions melt away into meaning- 
lessness. In particular the interior of the atom could 
not conceivably be explored by a gross survey. We 
cannot put a clock or a scale into the interior of an atom. 
It cannot be too strongly insisted that the terms dis- 
tance, period of time, mass, energy, momentum, etc., 
cannot be used in a description of an atom with the 
same meanings that they have in our gross experience. 
The atomic physicist who uses these terms must find 
his own meanings for them — must state the appliances 
which he requisitions when he imagines them to be 
measured. It is sometimes supposed that (in addition 
to electrical forces) there is a minute gravitational 
attraction between an atomic nucleus and the satellite 
electrons, obeying the same law as the gravitation 
between the sun and its planets. The supposition seems 
to me fantastic; but it is impossible to discuss it without 
any indication as to how the region within the atom is 
supposed to have been measured up. Apart from such 
measuring up the electron goes as it pleases "like the 
blessed gods". 

We have reached a point of great scientific and philo- 
sophic interest. The ten principal coefficients of cur- 
vature of the world are not strangers to us; they are 
already familiar in scientific discussion under other 
names (energy, momentum, stress). This is comparable 
with a famous turning-point in the development of elec- 
tromagnetic theory. The progress of the subject led to 
the consideration of waves of electric and magnetic force 
travelling through the aether; then it flashed upon 



156 GRAVITATION— THE EXPLANATION 

Maxwell that these waves were not strangers but were 
already familiar in our experience under the name of 
light. The method of identification is the same. It is 
calculated that electromagnetic waves will have just 
those properties which light is observed to have; so too 
it is calculated that the ten coefficients of curvature have 
just those properties which energy, momentum and stress 
are observed to have. We refer here to physical pro- 
perties only. No physical theory is expected to explain 
why there is a particular kind of image in our minds 
associated with light, nor why a conception of substance 
has arisen in our minds in connection with those parts 
of the world containing mass. 

This leads to a considerable simplification, because 
identity replaces causation. On the Newtonian theory 
no explanation of gravitation would be considered com- 
plete unless it described the mechanism by which a piece 
of matter gets a grip on the surrounding medium and 
makes it the carrier of the gravitational influence radi- 
ating from the matter. Nothing corresponding to this 
is required in the present theory. We do not ask how 
mass gets a grip on space-time and causes the curvature 
which our theory postulates. That would be as super- 
fluous as to ask how light gets a grip on the electro- 
magnetic medium so as to cause it to oscillate. The 
light is the oscillation; the mass is the curvature. There 
is no causal effect to be attributed to mass; still less is 
there any to be attributed to matter. The conception 
of matter, which we associate with these regions of un- 
usual contortion, is a monument erected by the mind to 
mark the scene of conflict. When you visit the site of a 
battle, do you ever ask how the monument that com- 
memorates it can have caused so much carnage? 

The philosophic outcome of this identification will 



NON-EUCLIDEAN GEOMETRY 157 

occupy us considerably in later chapters. Before leaving 
the subject of gravitation I wish to say a little about 
the meaning of space-curvature and non-Euclidean 
geometry. 

Non-Euclidean Geometry. I have been encouraging you 
to think of space-time as curved; but I have been careful 
to speak of this as a picture, not as a hypothesis. It is 
a graphical representation of the things we are talking 
about which supplies us with insight and guidance. 
What we glean from the picture can be expressed in a 
more non-committal way by saying that space-time has 
non-Euclidean geometry. The terms "curved space" 
and "non-Euclidean space" are used practically synony- 
mously; but they suggest rather different points of view. 
When we were trying to conceive finite and unbounded 
space (p. 81) the difficult step was the getting rid of 
the inside and the outside of the hypersphere. There is 
a similar step in the transition from curved space to 
non-Euclidean space — the dropping of all relations to 
an external (and imaginary) scaffolding and the holding 
on to those relations which exist within the space itself. 
If you ask what is the distance from Glasgow to New 
York there are two possible replies. One man will tell 
you the distance measured over the surface of the 
ocean; another will recollect that there is a still shorter 
distance by tunnel through the earth. The second man 
makes use of a dimension which the first had put out 
of mind. But if two men do not agree as to distances, 
they will not agree as to geometry; for geometry treats 
of the laws of distances. To forget or to be ignorant of 
a dimension lands us into a different geometry. Dis- 
tances for the second man obey a Euclidean geometry 
of three dimensions; distances for the first man obey 



158 GRAVITATION— THE EXPLANATION 

a non-Euclidean geometry of two dimensions. And so 
if you concentrate your attention on the earth's surface 
so hard that you forget that there is an inside or an 
outside to it, you will say that it is a two-dimensional 
manifold with non-Euclidean geometry; but if you 
recollect that there is three-dimensional space all round 
which affords shorter ways of getting from point to 
point, you can fly back to Euclid after all. You will then 
"explain away" the non-Euclidean geometry by saying 
that what you at first took for distances were not the 
proper distances. This seems to be the easiest way of 
seeing how a non-Euclidean geometry can arise — 
through mislaying a dimension — but we must not infer 
that non-Euclidean geometry is impossible unless it arises 
from this cause. 

In our four-dimensional world pervaded by gravitation 
the distances obey a non-Euclidean geometry. Is this 
because we are concentrating attention wholly on its 
four dimensions and have missed the short cuts through 
regions beyond? By the aid of six extra dimensions we 
can return to Euclidean geometry; in that case our usual 
distances from point to point in the world are not the 
"true" distances, the latter taking shorter routes through 
an eighth or ninth dimension. To bend the world in a 
super-world of ten dimensions so as to provide these 
short cuts does, I think, help us to form an idea 
of the properties of its non-Euclidean geometry; at any 
rate the picture suggests a useful vocabulary for de- 
scribing those properties. But we are not likely to accept 
these extra dimensions as a literal fact unless we regard 
non-Euclidean geometry as a thing which at all costs 
must be explained away. 

Of the two alternatives — a curved manifold in a 
Euclidean space of ten dimensions or a manifold with 



NON-EUCLIDEAN GEOMETRY 159 

non-Euclidean geometry and no extra dimensions — 
which is right? I would rather not attempt a direct 
answer, because I fear I should get lost in a fog of 
metaphysics. But I may say at once that I do not take 
the ten dimensions seriously; whereas I take the non- 
Euclidean geometry of the world very seriously, and 
I do not regard it as a thing which needs explaining 
away. The view, which some of us were taught at 
school, that the truth of Euclid's axioms can be seen in- 
tuitively, is universally rejected nowadays. We can no 
more settle the laws of space by intuition than we can 
settle the laws of heredity. If intuition is ruled out, the 
appeal must be to experiment — genuine open-minded ex- 
periment unfettered by any preconception as to what the 
verdict ought to be. We must not afterwards go back 
on the experiments because they make out space to be 
very slightly non-Euclidean. It is quite true that a way 
out could be found. By inventing extra dimensions we 
can make the non-Euclidean geometry of the world 
depend on a Euclidean geometry of ten dimensions; had 
the world proved to be Euclidean we could, I believe, 
have made its geometry depend on a non-Euclidean 
geometry of ten dimensions. No one would treat the 
latter suggestion seriously, and no reason can be given 
for treating the former more seriously. 

I do not think that the six extra dimensions have any 
stalwart defenders; but we. often meet with attempts to 
reimpose Euclidean geometry on the world in another 
way. The proposal, which is made quite unblushingly, 
is that since our measured lengths do not obey Euclidean 
geometry we must apply corrections to them — cook them 
— till they do. A closely related view often advocated 
is that space is neither Euclidean nor non-Euclidean; 
it is all a matter of convention and we are free to 



i6o GRAVITATION— THE EXPLANATION 

adopt any geometry we choose.* Naturally if we hold 
ourselves free to apply any correction we like to our 
experimental measures we can make them obey any 
law; but was it worth while saying this? The asser- 
tion that any kind of geometry is permissible could only 
be made on the assumption that lengths have no fixed 
value — that the physicist does not (or ought not to) 
mean anything in particular when he talks of length. 
I am afraid I shall have a difficulty in making my 
meaning clear to those who start from the assumption 
that my words mean nothing in particular; but for those 
who will accord them some meaning I will try to remove 
any possible doubt. The physicist is accustomed to state 
lengths to a great number of significant figures; to 
ascertain the significance of these lengths we must notice 
how they are derived; and we find that they are derived 
from a comparison with the extension of a standard of 
specified material constitution. (We may pause to notice 
that the extension of a standard material configuration 
may rightly be regarded as one of the earliest subjects 
of inquiry in a physical survey of our environment.) 
These lengths are a gateway through which knowledge 
of the world around us is sought. Whether or not they 
will remain prominent in the final picture of world- 
structure will transpire as the research proceeds; we do 
not prejudge that. Actually we soon find that space- 
lengths or time-lengths taken singly are relative, and only 

* As a recent illustration of this attitude I may refer to Bertrand 
Russell's Analysis of Matter, p. 78 — a book with which I do not often 
seriously disagree. "Whereas Eddington seems to regard it as necessary 
to adopt Einstein's variable space, Whitehead regards it as necessary 
to reject it. For my part, I do not see why we should agree with either 
view; the matter seems to be one of convenience in the interpretation of 
formulae." Russell's view is commended in a review by C. D. Broad. 
See also footnote, p. 142. 



NON-EUCLIDEAN GEOMETRY 161 

a combination of them could be expected to appear 
even in the humblest capacity in the ultimate world- 
structure. Meanwhile the first step through the gate- 
way takes us to the geometry obeyed by these lengths 
— very nearly Euclidean, but actually non-Euclidean and, 
as we have seen, a distinctive type of non-Euclidean 
geometry in which the ten principal coefficients of cur- 
vature vanish. We have shown in this chapter that 
the limitation is not arbitrary; it is a necessary property 
of lengths expressed in terms of the extension of a ma- 
terial standard, though it might have been surprising if 
it had occurred in lengths defined otherwise. Must we 
stop to notice the interjection that if we had meant 
something different by length we should have found a 
different geometry? Certainly we should; and if we 
had meant something different by electric force we should 
have found equations different from Maxwell's equations. 
Not only empirically but also by theoretical reasoning, 
we reach the geometry which we do because our lengths 
mean what they do. 

I have too long delayed dealing with the criticism of 
the pure mathematician who is under the impression 
that geometry is a subject that belongs entirely to him. 
Each branch of experimental knowledge tends to have 
associated with it a specialised body of mathematical 
investigations. The pure mathematician, at first called in 
as servant, presently likes to assert himself as master; 
the connexus of mathematical propositions becomes for 
him the main subject, and he does not ask permission 
from Nature when he wishes to vary or generalise the 
original premises. Thus he can arrive at a geometry 
unhampered by any restriction from actual space meas- 
ures; a potential theory unhampered by any question 
as to how gravitational and electrical potentials really 



1 62 GRAVITATION— THE EXPLANATION 

behave; a hydrodynamics of perfect fluids doing things 
which it would be contrary to the nature of any material 
fluid to do. But it seems to be only in geometry that 
he has forgotten that there ever was a physical subject 
of the same name, and even resents the application of 
the name to anything but his network of abstract math- 
ematics. I do not think it can be disputed that, both 
etymologically and traditionally, geometry is the science 
of measurement of the space around us; and however 
much the mathematical superstructure may now over- 
weigh the observational basis, it is properly speaking an 
experimental science. This is fully recognised in the 
"reformed" teaching of geometry in schools; boys are 
taught to verify by measurement that certain of the 
geometrical propositions are true or nearly true. No 
one questions the advantage of an unfettered develop- 
ment of geometry as a pure mathematical subject; but 
only in so far as this subject is linked to the quantities 
arising out of observation and measurement, will it find 
mention in a discussion of the Nature of the Physical 
World. 



Chapter VIII 

MAN'S PLACE IN THE UNIVERSE 

The Sidereal Universe. The largest telescopes reveal 
about a thousand million stars. Each increase in tele- 
scopic power adds to the number and we can scarcely 
set a limit to the multitude that must exist. Nevertheless 
there are signs of exhaustion, and it is clear that the 
distribution which surrounds us does not extend uni- 
formly through infinite space. At first an increase in 
light-grasp by one magnitude brings into view three 
times as many stars; but the factor diminishes so that 
at the limit of faintness reached by the giant telescopes 
a gain of one magnitude multiplies the number of stars 
seen by only 1.8, and the ratio at that stage is rapidly 
decreasing. It is as though we are approaching a limit 
at which increase of power will not bring into view very 
many additional stars. 

Attempts have been made to find the whole number 
of stars by a risky extrapolation of these counts, and 
totals ranging from 3000 to 30,000 millions are some- 
times quoted. But the difficulty is that the part of the 
stellar universe which we mainly survey is a local con- 
densation or star-cloud forming part of a much greater 
system. In certain directions in the sky our telescopes 
penetrate to the limits of the system, but in other direc- 
tions the extent is too great for us to fathom. The 
Milky Way, which on a dark night forms a gleaming 
belt round the sky, shows the direction in which there 
lie stars behind stars until vision fails. This great 
flattened distribution is called the Galactic System. It 
forms a disc of thickness small compared to its areal 

163 



164 MAN'S PLACE IN THE UNIVERSE 

extent. It is partly broken up into subordinate con- 
densations, which are probably coiled in spiral form like 
the spiral nebulae which are observed in great numbers 
in the heavens. The centre of the galactic system lies 
somewhere in the direction of the constellation Sagit- 
tarius; it is hidden from us not only by great distance but 
also to some extent by tracts of obscuring matter (dark 
nebulosity) which cuts off the light of the stars behind. 

We must distinguish then between our local star- 
cloud and the great galactic system of which it is a part. 
Mainly (but not exclusively) the star-counts relate to 
the local star-cloud, and it is this which the largest 
telescopes are beginning to exhaust. It too has a flat- 
tened form — flattened nearly in the same plane as the 
galactic system. If the galactic system is compared to 
a disc, the local star-cloud may be compared to a bun, 
its thickness being about one-third of its lateral ex- 
tension. Its size is such that light takes at least 2000 
years to cross from one side to the other; this measure- 
ment is necessarily rough because it relates to a vague 
condensation which is probably not sharply separated 
from other contiguous condensations. The extent of 
the whole spiral is of the order 100,000 light years. It 
can scarcely be doubted that the flattened form of the 
system is due to rapid rotation, and indeed there is 
direct evidence of strong rotational velocity; but it is 
one of the unexplained mysteries of evolution that 
nearly all celestial bodies have come to be endowed with 
fast rotation. 

Amid this great population the sun is a humble unit. 
It is a very ordinary star about midway in the scale of 
brilliancy. We know of stars which give at least 10,000 
times the light of the sun; we know also of stars which 
give 1/10,000 of its light. But those of inferior light 



THE SIDEREAL UNIVERSE 165 

greatly outnumber those of superior light. In mass, in 
surface temperature, in bulk, the sun belongs to a very 
common class of stars; its speed of motion is near the 
average; it shows none of the more conspicuous phe- 
nomena such as variability which excite the attention 
of astronomers. In the community of stars the sun 
corresponds to a respectable middle-class citizen. It 
happens to be quite near the centre of the local star- 
cloud; but this apparently favoured position is dis- 
counted by the fact that the star-cloud itself is placed 
very eccentrically in relation to the galactic system, being 
in fact near the confines of it. We cannot claim to be 
at the hub of the universe. 

The contemplation of the galaxy impresses us with 
the insignificance of our own little world; but we have 
to go still lower in the valley of humiliation. The 
galactic system is one among a million or more spiral 
nebulae. There seems now to be no doubt that, as has 
long been suspected, the spiral nebulae are "island uni- 
verses" detached from our own. They too are great 
systems of stars — or systems in the process of developing 
into stars — built on the same disc-like plan. We see 
some of them edgeways and can appreciate the flatness 
of the disc; others are broadside on and show the ar- 
rangement of the condensations in the form of a double 
spiral. Many show the effects of dark nebulosity 
breaking into the regularity -and blotting out the star- 
light. In a few of the nearest spirals it is possible to 
detect the brightest of the stars individually; variable 
stars and novae (or "new stars") are observed as in our 
own system. From the apparent magnitudes of the stars 
of recognisable character (especially the Cepheid vari- 
ables) it is possible to judge the distance. The nearest 
spiral nebula is 850,000 light years away. 



166 MAN'S PLACE IN THE UNIVERSE 

From the small amount of data yet collected it would 
seem that our own nebula or galactic system is ex- 
ceptionally large; it is even suggested that if the spiral 
nebulae are "islands" the galactic system is a "con- 
tinent". But we can scarcely venture to claim premier 
rank without much stronger evidence. At all events 
these other universes are aggregations of the order of 
ioo million stars. 

Again the question raises itself, How far does this 
distribution extend? Not the stars this time but uni- 
verses stretch one behind the other beyond sight. Does 
this distribution too come to an end? It may be that 
imagination must take another leap, envisaging super- 
systems which surpass the spiral nebulae as the spiral 
nebulae surpass the stars. But there is one feeble gleam 
of evidence that perhaps this time the summit of the 
hierarchy has been reached, and that the system of 
the spirals is actually the whole world. As has already 
been explained the modern view is that space is finite — 
finite though unbounded. In such a space light which 
has travelled an appreciable part of the way "round the 
world" is slowed down in its vibrations, with the result 
that all spectral lines are displaced towards the red. 
Ordinarily we interpret such a red displacement as sig- 
nifying receding velocity in the line of sight. Now it is 
a striking fact that a great majority of the spirals which 
have been measured show large receding velocities often 
exceeding iooo kilometres per second. There are only 
two serious exceptions, and these are the largest spirals 
which must be nearer to us than most of the others. 
On ordinary grounds it would be difficult to explain why 
these other universes should hurry away from us so fast 
and so unanimously. Why should they shun us like 
a plague? But the phenomenon is intelligible if what 



THE SCALE OF TIME 167 

has really been observed is the slowing down of vibra- 
tions consequent on the light from these objects having 
travelled an appreciable part of the way round the world. 
On that theory the radius of space is of the order twenty 
times the average distance of the nebulae observed, or 
say 100 million light years. That leaves room for a 
few million spirals; but there is nothing beyond. There 
is no beyond — in spherical space "beyond" brings us 
back towards the earth from the opposite direction.* 

The Scale of Time. The corridor of time stretches back 
through the past. We can have no conception how it 
all began. But at some stage we imagine the void to 
have been filled with matter rarified beyond the most 
tenuous nebula. The atoms sparsely strewn move hither 
and thither in formless disorder. 

Behold the throne 
Of Chaos and his dark pavilion spread 
Wide on the wasteful deep. 

Then slowly the power of gravitation is felt. Centres 
of condensation begin to establish themselves and draw 
in other matter. The first partitions are the star-systems 
such as our galactic system; sub-condensations separate 
the star-clouds or clusters; these divide again to give 
the stars. 

Evolution has not reached the same development in 

*A very much larger radius of space (io 11 light years) has recently 
been proposed by Hubble; but the basis of his calculation, though con- 
cerned with spiral nebulae, is different and to my mind unacceptable. 
It rests on an earlier theory of closed space proposed by Einstein which 
has generally been regarded as superseded. The theory given above (due 
to W. de Sitter) is, of course, very speculative, but it is the only clue we 
possess as to the dimensions of space. 



168 MAN'S PLACE IN THE UNIVERSE 

all parts. We observe nebulae and clusters in different 
stages of advance. Some stars are still highly diffuse; 
others are concentrated like the sun with density greater 
than water; others, still more advanced, have shrunk to 
unimaginable density. But no doubt can be entertained 
that the genesis of the stars is a single process of evolu- 
tion which has passed and is passing over a primordial 
distribution. Formerly it was freely speculated that the 
birth of a star was an individual event like the birth of 
an animal. From time to time two long extinct stars 
would collide and be turned into vapour by the energy of 
the collision; condensation would follow and life as a 
luminous body would begin all over again. We can 
scarcely affirm that this will never occur and that the 
sun is not destined to have a second or third innings; 
but it is clear from the various relations traced among 
the stars that the present stage of existence of the 
sidereal universe is the first innings. Groups of stars are 
found which move across the sky with common proper 
motion; these must have had a single origin and cannot 
have been formed by casual collisions. Another aban- 
doned speculation is that lucid stars may be the excep- 
tion, and that there may exist thousands of dead stars 
for every one that is seen shining. There are ways of 
estimating the total mass in interstellar space by its 
gravitational effect on the average speed of the stars; 
it is found that the lucid stars account for something 
approaching the total mass admissible and the amount 
left over for dark stars is very limited. 

Biologists and geologists carry back the history of the 
earth some thousand million years. Physical evidence 
based on the rate of transmutation of radioactive sub- 
stances seems to leave no escape from the conclusion 
that the older (Archaean) rocks in the earth's crust were 



PLURALITY OF WORLDS 169 

laid down 1200 million years ago. The sun must have 
been burning still longer, living (we now think) on its 
own matter which dissolves bit by bit into radiation. 
According to the theoretical time-scale, which seems 
best supported by astronomical evidence, the beginning 
of the sun as a luminous star must be dated five billion 
(5-I0 12 ) years ago. The theory which assigns this date 
cannot be trusted confidently, but it seems a reasonably 
safe conclusion that the sun's age does not exceed this 
limit. The future is not so restricted and the sun may 
continue as a star of increasing feebleness for 50 or 
500 billion years. The theory of sub-atomic energy 
has prolonged the life of a star from millions to bil- 
lions of years, and we may speculate on processes 
of rejuvenescence which might prolong the exist- 
ence of the sidereal universe from billions to trillions 
of years. But unless we can circumvent the second 
law of thermodynamics — which is as much as to 
say unless we can find cause for time to run back- 
wards — the ultimate decay draws surely nearer and the 
world will at the last come to a state of uniform 
changelessness. 

Does this prodigality of matter, of space, of time, 
find its culmination in Man? 

Plurality of Worlds. I will here put together the present 
astronomical evidence as tQ the habitability of other 
worlds. The popular idea that an answer to this ques- 
tion is one of the main aims of the study of celestial 
objects is rather disconcerting to the astronomer. Any- 
thing that he has to contribute is of the nature of frag- 
mentary hints picked up in the course of investigations 
with more practicable and commonplace purposes. 
Nevertheless, the mind is irresistibly drawn to play with 



170 MAN'S PLACE IN THE UNIVERSE 

the thought that somewhere in the universe there may 
be other beings "a little lower than the angels" whom 
Man may regard as his equals — or perhaps his 
superiors. 

It is idle to guess the forms that life might take in con- 
ditions differing from those of our planet. If I have 
rightly understood the view of palaeontologists, mam- 
malian life is the third terrestrial dynasty — Nature's 
third attempt to evolve an order of life sufficiently flex- 
ible to changing conditions and fitted to dominate the 
earth. Minor details in the balance of circumstances 
must greatly affect the possibility of life and the type of 
organism destined to prevail. Some critical branch- 
point in the course of evolution must be negotiated be- 
fore life can rise to the level of consciousness. All this 
is remote from the astronomer's line of study. To avoid 
endless conjecture I shall assume that the required con- 
ditions of habitability are not unlike those on the earth, 
and that if such conditions obtain life will automatically 
make its appearance. 

We survey first the planets of the solar system; of 
these only Venus and Mars seem at all eligible. Venus, 
so far as we know, would be well adapted for life 
similar to ours. It is about the same size as the earth, 
nearer the sun but probably not warmer, and it possesses 
an atmosphere of satisfactory density. Spectroscopic ob- 
servation has unexpectedly failed to give any indication of 
oxygen in the upper atmosphere and thus suggests a 
doubt as to whether free oxygen exists on the planet; 
but at present we hesitate to draw so definite an infer- 
ence. If transplanted to Venus we might perhaps con- 
tinue to live without much derangement of habit — 
except that I personally would have to find a new pro- 
fession, since Venus is not a good place for astronomers. 



PLURALITY OF WORLDS 171 

It is completely covered with cloud or mist. For this 
reason no definite surface markings can be made out, 
and it is still uncertain how fast it rotates on its axis 
and in which direction the axis lies. One curious theory 
may be mentioned though it should perhaps not be taken 
too seriously. It is thought by some that the great 
cavity occupied by the Pacific Ocean is a scar left by the 
moon when it was first disrupted from the earth. Evi- 
dently this cavity fulfils an important function in drain- 
ing away superfluous water, and if it were filled up 
practically all the continental area would be submerged. 
Thus indirectly the existence of dry land is bound up 
with the existence of the moon. But Venus has no moon, 
and since it seems to be similar to the earth in other 
respects, it may perhaps be inferred that it is a world 
which is all ocean — where fishes are supreme. The 
suggestion at any rate serves to remind us that the 
destinies of organic life may be determined by what 
are at first sight irrelevant accidents. 

The sun is an ordinary star and the earth is an 
ordinary planet, but the moon is not an ordinary satel- 
lite. No other known satellite is anything like so large 
in proportion to the planet which it attends. The moon 
contains about 1/80 part of the mass of the earth which 
seems a small ratio; but it is abnormally great compared 
with other satellites. The next highest ratio is found 
in the system of Saturn whose largest satellite Titan has 
1/4000 of the planet's mass. Very special circum- 
stances must have occurred in the history of the earth 
to have led to the breaking away of so unusual a frac- 
tion of the mass. The explanation proposed by Sir 
George Darwin, which is still regarded as most prob- 
able, is that a resonance in period occurred between 
the solar tides and the natural free period of vibration 



172 MAN'S PLACE IN THE UNIVERSE 

of the globe of the earth. The tidal deformation of the 
earth thus grew to large amplitude, ending in a cata- 
clysm which separated the great lump of material that 
formed the moon. Other planets escaped this dangerous 
coincidence of period, and their satellites separated by- 
more normal development. If ever I meet a being who 
has lived in another world, I shall feel very humble in 
most respects, but I expect to be able to boast a little 
about the moon. 

Mars is the only planet whose solid surface can be 
seen and studied; and it tempts us to consider the possi- 
bility of life in more detail. Its smaller size leads to 
considerably different conditions; but the two essentials, 
air and water, are both present though scanty. The 
Martian atmosphere is thinner than our own but it is 
perhaps adequate. It has been proved to contain oxy- 
gen. There is no ocean; the surface markings repre- 
sent, not sea and land, but red desert and darker ground 
which is perhaps moist and fertile. A conspicuous fea- 
ture is the white cap covering the pole which is clearly 
a deposit of snow; it must be quite shallow since it melts 
away completely in the summer. Photographs show 
from time to time indubitable clouds which blot out 
temporarily large areas of surface detail; clear weather, 
however, is more usual. The air, if cloudless, is slightly 
hazy. W. H. Wright has shown this very convincingly 
by comparing photographs taken with light of dif- 
ferent wave-lengths. Light of short wave-length is 
much scattered by haze and accordingly the ordinary 
photographs are disappointingly blurry. Much sharper 
surface-detail is shown when visual yellow light is 
employed (a yellow screen being commonly used to 
adapt visual telescopes for photography) ; being of 
longer wave-length the visual rays penetrate the haze 



PLURALITY OF WORLDS 173 

more easily.* Still clearer detail is obtained by photo- 
graphing with the long infra-red waves. 

Great attention has lately been paid to the deter- 
mination of the temperature of the surface of Mars; it 
is possible to find this by direct measurement of the heat 
rediated to us from different parts of the surface. The 
results, though in many respects informative, are 
scarcely accurate and accordant enough to give a defi- 
nite idea of the climatology. Naturally the tempera- 
ture varies a great deal between day and night and in 
different latitudes; but on the average the conditions 
are decidedly chilly. Even at the equator the tempera- 
ture falls below freezing point at sunset. If we accepted 
the present determinations as definitive we should have 
some doubt as to whether life could endure the con- 
ditions. 

In one of Huxley's Essays there occurs the passage 
"Until human life is longer and the duties of the 
present press less heavily I do not think that wise men 
will occupy themselves with Jovian or Martian natural 
history." To-day it would seem that Martian natural 
history is not altogether beyond the limits of serious 
science. At least the surface of Mars shows a seasonal 
change such as we might well imagine the forest-clad 
earth would show to an outside onlooker. This seasonal 
change of appearance is very conspicuous to the atten- 
tive observer. As the spring in one hemisphere advances 
(I mean, of course, the Martian spring), the darker 
areas, which are at first few and faint, extend and 
deepen in contrast. The same regions darken year after 

* It seems to have been a fortunate circumstance that the pioneers 
of Martian photography had no suitable photographic telescopes and 
had to adapt visual telescopes — thus employing visual (yellow) light 
which, as it turned out, was essential for good results. 



i 7 4 MAN'S PLACE IN THE UNIVERSE 

year at nearly the same date in the Martian calendar. 
It may be that there is an inorganic explanation; the 
spring rains moisten the surface and change its colour. 
But it is perhaps unlikely that there is enough rain 
to bring about this change as a direct effect. It is 
easier to believe that we are witnessing the annual 
awakening of vegetation so familiar on our own 
planet. 

The existence of oxygen in the Martian atmosphere 
supplies another argument in support of the existence 
of vegetable life. Oxygen combines freely with many 
elements, and the rocks in the earth's crust are thirsty 
for oxygen. They would in course of time bring about 
its complete disappearance from the air, were it not that 
the vegetation extracts it from the soil and sets it free 
again. If oxygen in the terrestrial atmosphere is main- 
tained in this way, it would seem reasonable to assume 
that vegetable life is required to play the same part on 
Mars. Taking this in conjunction with the evidence of 
the seasonal changes of appearance, a rather strong case 
for the existence of vegetation seems to have been made 
out. 

If vegetable life must be admitted, can we exclude 
animal life? I have come to the end of the astronomical 
data and can take no responsibility for anything further 
that you may infer. It is true that the late Prof. Lowell 
argued that certain more or less straight markings on 
the planet represent an artificial irrigation system and 
are the signs of an advanced civilisation; but this theory 
has not, I think, won much support. In justice to the 
author of this speculation it should be said that his own 
work and that of his observatory have made a magni- 
ficent contribution to our knowledge of Mars; but few 
would follow him all the way on the more picturesque 



FORMATION OF PLANETARY SYSTEMS 175 

side of his conclusions.* Finally we may stress one 
point. Mars has every appearance of being a planet 
long past its prime; and it is in any case improbable that 
two planets differing so much as Mars and the Earth 
would be in the zenith of biological development con- 
temporaneously. 

Formation of Planetary Systems. If the planets of the 
solar system should fail us, there remain some thousands 
of millions of stars which we have been accustomed to 
regard as suns ruling attendant systems of planets. It 
has seemed a presumption, bordering almost on impiety, 
to deny to them life of the same order of creation as 
ourselves. It would indeed be rash to assume that 
nowhere else in the universe has Nature repeated the 
strange experiment which she has performed on the 
earth. But there are considerations which must hold us 
back from populating the universe too liberally. 

On examining the stars with a telescope we are sur- 
prised to find how many of those which appear single 
points to the eye are actually two stars close together. 
When the telescope fails to separate them the spectro- 
scope often reveals two stars in orbital revolution round 
each other. At least one star in three is double — a pair 
of self-luminous globes both comparable in dimensions 
with the sun. The single supreme sun is accordingly 
not the only product of evolution; not much less fre- 
quently the development has taken another turn and 
resulted in two suns closely associated. We may prob- 
ably rule out the possibility of planets in double stars. 

♦Mars is not seen under favourable conditions except from low lati- 
tudes and high altitudes. Astronomers who have not these advantages 
are reluctant to form a decided opinion on the many controversial points 
that have arisen. 



176 MAN'S PLACE IN THE UNIVERSE 

Not only is there a difficulty in ascribing to them per- 
manent orbits under the more complicated field of gravi- 
tation, but a cause for the formation of planets seems 
to be lacking. The star has satisfied its impulse to 
fission in another manner; it has divided into two nearly 
equal portions instead of throwing off a succession of 
tiny fragments. 

The most obvious cause of division is excessive rota- 
tion. As the gaseous globe contracts it spins fast and 
faster until a time may come when it can no longer hold 
together, and some kind of relief must be found. Ac- 
cording to the nebular hypothesis of Laplace the sun 
gained relief by throwing off successively rings of matter 
which have formed the planets. But were it not for 
this one instance of a planetary system which is known 
to us, we should have concluded from the thousands of 
double stars in the sky that the common consequence of 
excessive rotation is to divide the star into two bodies 
of equal rank. 

It might still be held that the ejection of a planetary 
system and the fission into a double star are alternative 
solutions of the problem arising from excessive rotation, 
the star taking one course or the other according to 
circumstances. We know of myriads of double stars 
and of only one planetary system; but in any case it is 
beyond our power to detect other planetary systems if 
they exist. We can only appeal to the results of theo- 
retical study of rotating masses of gas; the work pre- 
sents many complications and the results may not be 
final; but the researches of Sir J. H. Jeans lead to the 
conclusion that rotational break-up produces a double 
star and never a system of planets. The solar system is 
not the typical product of development of a star; it is 
not even a common variety of development; it is a freak. 



FORMATION OF PLANETARY SYSTEMS 177 

By elimination of alternatives it appears that a con- 
figuration resembling the solar system would only be 
formed if at a certain stage of condensation an unusual 
accident had occurred. According to Jeans the accident 
was the close approach of another star casually pursuing 
its way through space. This star must have passed 
within a distance not far outside the orbit of Neptune; 
it must not have passed too rapidly, but have slowly 
overtaken or been overtaken by the sun. By tidal dis- 
tortion it raised big protuberances on the sun, and caused 
it to spurt out filaments of matter which have condensed 
to form the planets. That was more than a thousand 
million years ago. The intruding star has since gone on 
its way and mingled with the others; its legacy of a 
system of planets remains, including a globe habitable 
by man. 

Even in the long life of a star encounters of this kind 
must be extremely rare. The density of distribution of 
stars in space has been compared to that of twenty 
tennis-balls roaming the whole interior of the earth. 
The accident that gave birth to the solar system may be 
compared to the casual approach of two of these balls 
within a few yards of one another. The data are too 
vague to give any definite estimate of the odds against 
this occurence, but I should judge that perhaps not one 
in a hundred millions of stars can have undergone this 
experience in the right stage and conditions to result in 
the formation of a system of planets. 

However doubtful this conclusion as to the rarity of 
solar systems may be, it is a useful corrective to the view 
too facilely adopted which looks upon every star as a 
likely minister 'to life. We know the prodigality of 
Nature. How many acorns are scattered for one that 
grows to an oak? And need she be more careful of her 



i;8 MAN'S PLACE IN THE UNIVERSE 

stars than of her acorns? If indeed she has no grander 
aim than to provide a home for her greatest experiment, 
Man, it would be just like her methods to scatter a mil- 
lion stars whereof one might haply achieve her purpose. 

The number of possible abodes of life severely 
restricted in this way at the outset may no doubt be 
winnowed down further. On our house-hunting expedi- 
tion we shall find it necessary to reject many apparently 
eligible mansions on points of detail. Trivial circum- 
stances may decide whether organic forms originate at 
all; further conditions may decide whether life ascends 
to a complexity like ours or remains in a lower form. 
I presume, however, that at the end of the weeding 
out there will be left a few rival earths dotted here and 
there about the universe. 

A further point arises if we have especially in mind 
contemporaneous life. The time during which man has 
been on the earth is extremely small compared with the 
age of the earth or of the sun. There is no obvious 
physical reason why, having once arrived, man should 
not continue to populate the earth for another ten billion 
years or so; but — well, can you contemplate it? Assum- 
ing that the stage of highly developed life is a very 
small fraction of the inorganic history of the star, the 
rival earths are in general places where conscious life 
has already vanished or is yet to come. I do not think 
that the whole purpose of the Creation has been staked 
on the one planet where we live; and in the long run we 
cannot deem ourselves the only race that has been or 
will be gifted with the mystery of consciousness. But 
I feel inclined to claim that at the present time our race 
is supreme; and not one of the profusion of stars in 
their myriad clusters looks down on scenes comparable 
to those which are passing beneath the rays of the sun. 



Chapter IX 
THE QUANTUM THEORY 

The Origin of the Trouble. Nowadays whenever en- 
thusiasts meet together to discuss theoretical physics the 
talk sooner or later turns in a certain direction. You 
leave them conversing on their special problems or the 
latest discoveries; but return after an hour and it is any 
odds that they will have reached an all-engrossing topic 
— the desperate state of their ignorance. This is not a 
pose. It is not even scientific modesty, because the atti- 
tude is often one of naive surprise that Nature should 
have hidden her fundamental secret successfully from 
such powerful intellects as ours. It is simply that we 
have turned a corner in the path of progress and our 
ignorance stands revealed before us, appalling and insist- 
ent. There is something radically wrong with the pres- 
ent fundamental conceptions of physics and we do not 
see how to set it right. 

The cause of all this trouble is a little thing called h 
which crops up continually in a wide range of experi- 
ments. In one sense we know just what h is, because 
there are a variety of ways of measuring it; h is 

.0000000000000000000000000065 5 erg-seconds. 

That will (rightly) suggest to you that h is something 
very small; but the most important information is con- 
tained in the concluding phrase erg-seconds. The erg 
is the unit of energy and the second is the unit of time; 
so that we learn that h is of the nature of energy multi- 
plied by time. 

Now in practical life it does not often occur to us to 

179 



180 THE QUANTUM THEORY 

multiply energy by time. We often divide energy by 
time. For example, the motorist divides the output of 
energy of his engine by time and so obtains the horse- 
power. Conversely an electric supply company multi- 
plies the horse-power or kilowatts by the number of 
hours of consumption and sends in its bill accordingly. 
But to multiply by hours again would seem a very odd 
sort of thing to do. 

But it does not seem quite so strange when we look 
at it in the absolute four-dimensional world. Quantities 
such as energy, which we think of as existing at an 
instant, belong to three-dimensional space, and they need 
to be multiplied by a duration to give them a thickness 
before they can be put into the four-dimensional world. 
Consider a portion of space, say Great Britain; we 
should describe the amount of humanity in it as 40 
million men. But consider a portion of space-time, say 
Great Britain between 19 15 and 1925; we must describe 
the amount of humanity in it as 400 million man-years. 
To describe the human content of the world from a 
space-time point of view we have to take a unit which is 
limited not only in space but in time. Similarly if some 
other kind of content of space is described as so many 
ergs, the corresponding content of a region of space-time 
will be described as so many erg-seconds. 

We call this quantity in the four-dimensional world 
which is the analogue or adaptation of energy in the 
three-dimensional world by the technical name action. The 
name does not seem to have any special appropriateness, 
but we have to accept it. Erg-seconds or action belongs 
to Minkowski's world which is common to all observers, 
and so it is absolute. It is one of the very few absolute 
quantities noticed in pre-relativity physics. Except for 
action and entropy (which belongs to an entirely different 



THE ORIGIN OF THE TROUBLE 181 

class of physical conceptions) all the quantities promi- 
nent in pre-relativity physics refer to the three-dimen- 
sional sections which are different for different observers. 

Long before the theory of relativity showed us that 
action was likely to have a special importance in the 
scheme of Nature on account of its absoluteness, long 
before the particular piece of action h began to turn up 
in experiments, the investigators of theoretical dynamics 
were making great use of action. It was especially the 
work of Sir William Hamilton which brought it to the 
fore; and since then very extensive theoretical develop- 
ments of dynamics have been made on this basis. I 
need only refer to the standard treatise on Analytical 
Dynamics by your own (Edinburgh) Professor*, which 
fairly reeks of it. It was not difficult to appreciate the 
fundamental importance and significance of the main 
principle; but it must be confessed that to the non- 
specialist the interest of the more elaborate develop- 
ments did not seem very obvious — except as an ingenious 
way of making easy things difficult. In the end the 
instinct which led to these researches has justified itself 
emphatically. To follow any of the progress in the 
quantum theory of the atom since about 19 17, it is 
necessary to have plunged rather deeply into the Hamil- 
tonian theory of dynamics. It is remarkable that just 
as Einstein found ready prepared by the mathematicians 
the Tensor Calculus which he needed for developing 
his great theory of gravitation, so the quantum physicists 
found ready for them an extensive action-theory of 
dynamics without which they could not have made head- 
way. 

But neither the absolute importance of action in the 
four-dimensional world, nor its earlier prominence in 

* Prof. E. T. Whittaker. 



1 82 THE QUANTUM THEORY 

Hamiltonian dynamics, prepares us for the discovery 
that a particular lump of it can have a special import- 
ance. And yet a lump of standard size 6-55. io~ 27 erg- 
seconds is continually turning up experimentally. It is 
all very well to say that we must think of action as 
atomic and regard this lump as the atom of action. We 
cannot do it. We have been trying hard for the last 
ten years. Our present picture of the world shows 
action in a form quite incompatible with this kind of 
atomic structure, and the picture will have to be redrawn. 
There must in fact be a radical change in the funda- 
mental conceptions on which our scheme of physics is 
founded; the problem is to discover the particular 
change required. Since 1925 new ideas have been 
brought into the subject which seem to make the dead- 
lock less complete, and give us an inkling of the nature 
of the revolution that must come; but there has been 
no general solution of the difficulty. The new ideas will 
be the subject of the next chapter. Here it seems best 
to limit ourselves to the standpoint of 1925, except at 
the very end of the chapter, where we prepare for the 
transition. 

The Atom of Action. Remembering that action has two 
ingredients, namely, energy and time, we must look about 
in Nature for a definite quantity of energy with which 
there is associated some definite period of time. That is 
the way in which without artificial section a particular 
lump of action can be separated from the rest of the action 
which fills the universe. For example, the energy of consti- 
tution of an electron is a definite and known quantity; it 
is an aggregation of energy which occurs naturally in all 
parts of the universe. But there is no particular duration 
of time associated with it that we are aware of, and so it 



THE ATOM OF ACTION 183 

does not suggest to us any particular lump of action. 
We must turn to a form of energy which has a definite 
and discoverable period of time associated with it, such 
as a train of light-waves; these carry with them a unit 
of time, namely, the period of their vibration. The 
yellow light from sodium consists of aethereal vibrations 
of period 510 billions to the second. At first sight we 
seem to be faced with the converse difficulty; we have 
now our definite period of time; but how are we to cut 
up into natural units the energy coming from a sodium 
flame? We should, of course, single out the light pro- 
ceeding from a single atom, but this will not break up 
into units unless the atom emits light discontinuously. 

It turns out that the atom does emit light discontin- 
uously. It sends out a long train of waves and then 
stops. It has to be restarted by some kind of stimula- 
tion before it emits again. We do not perceive this 
intermittence in an ordinary beam of light, because there 
are myriads of atoms engaged in the production. 

The amount of energy coming away from the sodium 
atom during any one of these discontinuous emissions 
is found to be 3-4. io -12 ergs. This energy is, as we 
have seen, marked by a distinctive period 1-9. io~ 15 sees. 
We have thus the two ingredients necessary for a 
natural lump of action. Multiply them together, and 
we obtain 6-55. io~ 27 erg-seconds. That is the quan- 
tity h. 

The remarkable law of Nature is that we are con- 
tinually getting the same numerical results. We may 
take another source of light — hydrogen, calcium, or any 
other atom. The energy will be a different number of 
ergs; the period will be a different number of seconds; 
but the product will be the same number of erg-seconds. 
The same applies to X-rays, to gamma rays and to other 



1 84 THE QUANTUM THEORY 

forms of radiation. It applies to light absorbed by an 
atom as well as to light emitted, the absorption being 
discontinuous also. Evidently h is a kind of atom — 
something which coheres as one unit in the processes of 
radiation; it is not an atom of matter but an atom or, 
as we usually call it, a quantum of the more elusive 
entity action. Whereas there are 92 different kinds of 
material atoms there is only one quantum of action — 
the same whatever the material it is associated with. 
I say the same without reservation. You might perhaps 
think that there must be some qualitative difference 
between the quantum of red light and the quantum of 
blue light, although both contain the same number of 
erg-seconds; but the apparent difference is only relative 
to a frame of space and time and does not concern the 
absolute lump of action. By approaching the light- 
source at high speed we change the red light to blue 
light in accordance with Doppler's principle; the energy 
of the waves is also changed by being referred to a 
new frame of reference. A sodium flame and a hydro- 
gen flame are throwing out at us the same lumps of 
action, only these lumps are rather differently orientated 
with respect to the Now lines which we have drawn 
across the four-dimensional world. If we change our 
motion so as to alter the direction of the Now lines, 
we can see the lumps of sodium origin under the same 
orientation in which we formerly saw the lumps of 
hydrogen origin and recognise that they are actually the 
same. 

We noticed in chapter iv that the shuffling of energy 
can become complete, so that a definite state is reached 
known as thermodynamical equilibrium; and we re- 
marked that this is only possible if indivisible units are 
being shuffled. If the cards can be torn into smaller and 



CONFLICT WITH WAVE-THEORY 185 

smaller pieces without limit there is no end to the 
process of shuffling. The indivisible units in the shuf- 
fling of energy are the quanta. By radiation absorp- 
tion and scattering energy is shuffled among the different 
receptacles in matter and aether, but only a whole 
quantum passes at each step. It was in fact this definite- 
ness of thermodynamical equilibrium which first put 
Prof. Max Planck on the track of the quantum; and the 
magnitude of h was first calculated by analysis of the 
observed composition of the radiation in the final state 
of randomness. Progress of the theory in its adolescent 
stage was largely due to Einstein so far as concerns the 
general principles and to Bohr as regards its connection 
with atomic structure. 

The paradoxical nature of the quantum is that 
although it is indivisible it does not hang together. We 
examined first a case in which a quantity of energy was 
obviously cohering together, viz. an electron, but we did 
not find h; then we turned our attention to a case in which 
the energy was obviously dissolving away through space, 
viz. light-waves, and immediately h appeared. The 
atom of action seems to have no coherence in space; 
it has a unity which overleaps space. How can such a 
unity be made to appear in our picture of a world 
extended through space and time? 

Conflict with the Wave-Theory of Light. The pursuit of 
the quantum leads to many surprises; but probably none 
is more outrageous to our preconceptions than the 
regathering of light and other radiant energy into 
A-units, when all the classical pictures show it to be 
dispersing more and more. Consider the light-waves 
which are the result of a single emission by a single atom 
on the star Sirius. These bear away a certain amount of 



1 86 THE QUANTUM THEORY 

energy endowed with a certain period, and the product 
of the two is //. The period is carried by the waves 
without change, but the energy spreads out in an ever- 
widening circle. Eight years and nine months after the 
emission the wave-front is due to reach the earth. A 
few minutes before the arrival some person takes it into 
his head to go out and admire the glories of the heavens 
and — in short — to stick his eye in the way. The light- 
waves when they started could have had no notion what 
they were going to hit; for all they knew they were 
bound on a journey through endless space, as most of 
their colleagues were. Their energy would seem to be 
dissipated beyond recovery over a sphere of 50 billion 
miles' radius. And yet if that energy is ever to enter 
matter again, if it is to work those chemical changes in 
the retina which give rise to the sensation of light, it 
must enter as a single quantum of action h. Just 
6-55. 1 o -27 erg-seconds must enter or none at all. Just 
as the emitting atom regardless of all laws of classical 
physics is determined that whatever goes out of it shall 
be just /*, so the receiving atom is determined that what- 
ever comes into it shall be just h. Not all the light- 
waves pass by without entering the eye; for somehow 
we are able to see Sirius. How is it managed? Do the 
ripples striking the eye send a message round to the 
back part of the wave, saying, "We have found an eye. 
Let's all crowd into it!" 

Attempts to account for this phenomenon follow two 
main devices which we may describe as the "collection- 
box" theory and the "sweepstake" theory, respectively. 
Making no effort to translate them into scientific 
language, they amount to this: In the first the atom 
holds a collection-box into which each arriving group 
of waves pays a very small contribution; when the 



CONFLICT WITH WAVE-THEORY 187 

amount in the box reaches a whole quantum, it enters 
the atom. In the second the atom uses the small frac- 
tion of a quantum offered to it to buy a ticket in a 
sweepstake in which the prizes are whole quanta; some 
of the atoms will win whole quanta which they can 
absorb, and it is these winning atoms in our retina 
which tell us of the existence of Sirius. 

The collection-box explanation is not tenable. As 
Jeans once said, not only does the quantum theory 
forbid us to kill two birds with one stone; it will not 
even let us kill one bird with two stones. I cannot go 
fully into the reasons against this theory, but may 
illustrate one or two of the difficulties. One serious 
difficulty would arise from the half-filled collection- 
boxes. We shall see this more easily if, instead of 
atoms, we consider molecules which also absorb only 
full quanta. A molecule might begin to collect the 
various kinds of light which it can absorb, but before it 
has collected a quantum of any one kind it takes part 
in a chemical reaction. New compounds are formed 
which no longer absorb the old kinds of light; they have 
entirely different absorption spectra. They would have 
to start afresh to collect the corresponding kinds of 
light. What is to be done with the old accumulations 
now useless, since they can never be completed? One 
thing is certain; they are not tipped out into the aether 
when the chemical change occurs. 

A phenomenon which seems directly opposed to any 
kind of collection-box explanation is the photoelectric 
effect. When light shines on metallic films of sodium, 
potassium, rubidium, etc., free electrons are discharged 
from the film. They fly away at high speed, and it is 
possible to measure experimentally their speed or 
energy. Undoubtedly it is the incident light which 



188 THE QUANTUM THEORY 

provides the energy of these explosions, but the phe- 
nomenon is governed by a remarkable rule. Firstly, the 
speed of the electrons is not increased by using more 
powerful light. Concentration of the light produces 
more explosions but not more powerful explosions. 
Secondly, the speed is increased by using bluer light, i.e. 
light of shorter period. For example, the feeble light 
reaching us from Sirius will cause more powerful ejec- 
tions of electrons than full sunlight, because Sirius is 
bluer than the sun; the remoteness of Sirius does not 
weaken the ejections though it reduces their number. 

This is a straightforward quantum phenomenon. 
Every electron flying out of the metal has picked up just 
one quantum from the incident light. Since the /z-rule 
associates the greater energy with the shorter vibration 
period, bluer light gives the more intense energy. 
Experiments show that (after deducting a constant 
"threshold" energy used up in extricating the electron 
from the film) each electron comes out with a kinetic 
energy equal to the energy of the quantum of incident 
light. 

The film can be prepared in the dark; but on ex- 
posure to feeble light electrons immediately begin to 
fly out before any of the collection-boxes could have 
been filled by fair means. Nor can we appeal to any 
trigger action of the light releasing an electron already 
loaded up with energy for its journey; it is the nature 
of the light which settles the amount of the load. The 
light calls the tune, therefore the light must pay the 
piper. Only classical theory does not provide light with 
a pocket to pay from. 

It is always difficult to make a fence of objections so 
thorough as to rule out all progress along a certain line 
of explanation. But even if it is still possible to wriggle 



CONFLICT WITH WAVE-THEORY 189 

on, there comes a time when one begins to perceive that 
the evasions are far-fetched. If we have any instinct 
that can recognise a fundamental law of Nature when 
it sees one, that instinct tells us that the interaction of 
radiation and matter in single quanta is something lying 
at the root of world-structure and not a casual detail in 
the mechanism of the atom. Accordingly we turn to the 
"sweepstake" theory, which sees in this phenomenon a 
starting-point for a radical revision of the classical con- 
ceptions. 

Suppose that the light-waves are of such intensity that, 
according to the usual reckoning of their energy, one- 
millionth of a quantum is brought within range of each 
atom. The unexpected phenomenon is that instead of 
each atom absorbing one-millionth of a quantum, one 
atom out of every million absorbs a whole quantum. 
That whole quanta are absorbed is shown by the photo- 
electric experiments already described, since each of the 
issuing electrons has managed to secure the energy of a 
whole quantum. 

It would seem that what the light-waves were really 
bearing within reach of each atom was not a millionth 
of a quantum but a millionth chance of securing a whole 
quantum. The wave-theory of light pictures and 
describes something evenly distributed over the whole 
wave-front which has usually been identified with energy. 
Owing to well-established phenomena such as interfer- 
ence and diffraction it seems impossible to deny this uni- 
formity, but we must give it another interpretation; it 
is a uniform chance of energy. Following the rather 
old-fashioned definition of energy as "capacity for doing 
work" the waves carry over their whole front a uniform 
chance of doing work. It is the propagation of a chance 
which the wave-theory studies. 



190 THE QUANTUM THEORY 

Different views may be held as to how the prize- 
drawing is conducted on the sweepstake theory. Some 
hold that the lucky part of the wave-front is already 
marked before the atom is reached. In addition to the 
propagation of uniform waves the propagation of a 
photon or "ray of luck" is involved. This seems to me 
out of keeping with the general trend of the modern 
quantum theory; and although most authorities now take 
this view, which is said to be indicated definitely by 
certain experiments, I do not place much reliance on the 
stability of this opinion. 

Theory of the Atom. We return now to further experi- 
mental knowledge of quanta. The mysterious quantity 
h crops up inside the atom as well as outside it. Let us 
take the simplest of all atoms, namely, the hydrogen 
atom. This consists of a proton and an electron, that is 
to say a unit charge of positive electricity and a unit 
charge of negative electricity. The proton carries nearly 
all the mass of the atom and remains rock-like at the 
centre, whilst the nimble electron moves round in a 
circular or elliptic orbit under the inverse square-law 
of attraction between them. The system is thus very 
like a sun and a planet. But whereas in the solar system 
the planet's orbit may be of any size and any eccentricity, 
the electron's orbit is restricted to a definite series of 
sizes and shapes. There is nothing in the classical 
theory of electromagnetism to impose such a restriction; 
but the restriction exists, and the law imposing it has 
been discovered. It arises because the atom is arranging 
to make something in its interior equal to h. The inter- 
mediate orbits are excluded because they would involve 
fractions of /*, and h cannot be divided. 

But there is one relaxation. When wave-energy is 



THEORY OF THE ATOM 191 

sent out from or taken into the atom, the amount and 
period must correspond exactly to h. But as regards 
its internal arrangements the atom has no objection to 
2/*, 3/*, 4/i, etc.; it only insists that fractions shall be 
excluded. That is why there are many alternative orbits 
for the electron corresponding to different integral mul- 
tipliers of h. We call these multipliers quantum num- 
bers, and speak of 1 -quantum orbits, 2-quantum orbits, 
etc. I will not enter here into the exact definition of 
what it is that has to be an exact multiple of h; but it 
is something which, viewed in the four-dimensional 
world, is at once seen to be action though this may not 
be so apparent when we view it in the ordinary way in 
three-dimensional sections. Also several features of the 
atom are regulated independently by this rule, and 
accordingly there are several quantum numbers — one for 
each feature; but to avoid technical complication I shall 
refer only to the quantum numbers belonging to one 
leading feature. 

According to this picture of the atom, which is due 
to Niels Bohr, the only possible change of state is the 
transfer of an electron from one quantum orbit to 
another. Such a jump must occur whenever light is 
absorbed or emitted. Suppose then that an electron which 
has been travelling in one of the higher orbits jumps 
down into an orbit of less energy. The atom will then 
have a certain amount of surplus energy that must be got 
rid of. The lump of energy is fixed, and it remains to 
settle the period of vibration that it shall have when it 
changes into aether-waves. It seems incredible that the 
atom should get hold of the aether and shake it in any 
other period than one of those in which it is itself 
vibrating. Yet it is the experimental fact that, when the 
atom by radiating sets the aether in vibration, the 



192 THE QUANTUM THEORY 

periods of its electronic circulation are ignored and the 
period of the aether-waves is settled not by any pictur- 
able mechanism but by the seemingly artificial h-rulc. It 
would seem that the atom carelessly throws overboard 
a lump of energy which, as it glides into the aether, 
moulds itself into a quantum of action by taking on the 
period required to make the product of energy and 
period equal to h. If this unmechanical process of emis- 
sion seems contrary to our preconceptions, the exactly 
converse process of absorption is even more so. Here 
the atom has to look out for a lump of energy of the 
exact amount required to raise an electron to the higher 
orbit. It can only extract such a lump from aether- 
waves of particular period — not a period which has 
resonance with the structure of the atom, but the period 
which makes the energy into an exact quantum. 

As the adjustment between the energy of the orbit jump 
and the period of the light carrying away that energy so 
as to give the constant quantity h is perhaps the most 
striking evidence of the dominance of the quantum, it 
will be worth while to explain how the energy of an 
orbit jump in an atom can be measured. It is possible to 
impart to a single electron a known amount of energy by 
making it travel along an electric field with a measured 
drop of potential. If this projectile hits an atom it may 
cause one of the electrons circulating in the atom to 
jump to an upper orbit, but, of course, only if its energy 
is sufficient to supply that required for the jump; if the 
electron has too little energy it can do nothing and must 
pass on with its energy intact. Let us fire a stream of 
electrons all endowed with the same known energy 
into the midst of a group of atoms. If the energy is 
below that corresponding to an orbit jump, the stream 
will pass through without interference other than 



CLASSICAL AND QUANTUM LAWS 193 

ordinary scattering. Now gradually increase the energy 
of the electrons; quite suddenly we find that the electrons 
are leaving a great deal of their energy behind. That 
means that the critical energy has been reached and 
orbit jumps are being excited. Thus we have a means 
of measuring the critical energy which is just that of the 
jump — the difference of energy of the two states of the 
atom. This method of measurement has the advantage 
that it does not involve any knowledge of the constant h, 
so that there is no fear of a vicious circle when we use 
the measured energies to test the h rule.* Incidentally 
this experiment provides another argument against the 
collection-box theory. Small contributions of energy are 
not thankfully received, and electrons which offer any- 
thing less than the full contribution for a jump are not 
allowed to make any payment at all. 

Relation of Classical Laws to Quantum Laws. To fol- 
low up the verification and successful application of the 
quantum laws would lead to a detailed survey of the 
greater part of modern physics — specific heats, mag- 
netism, X-rays, radioactivity, and so on. We must leave 
this and return to a general consideration of the rela- 
tion between classical laws and quantum laws. For at 
least fifteen years we have used classical laws and quan- 
tum laws alongside one another notwithstanding the 
irreconcilability of their conceptions. In the model atom 
the electrons are supposed to traverse their orbits under 
the classical laws of electrodynamics; but they jump 
from one orbit to another in a way entirely incon- 
sistent with those laws. The energies of the orbits 

* Since the h rule is now well established the energies of different 
states of the atoms are usually calculated by its aid; to use these to test 
the rule would be a vicious circle. 



194 THE QUANTUM THEORY 

in hydrogen are calculated by classical laws; but one 
of the purposes of the calculation is to verify the 
association of energy and period in the unit /*, which is 
contrary to classical laws of radiation. The whole 
procedure is glaringly contradictory but conspicuously 
successful. 

In my observatory there is a telescope which con- 
denses the light of a star on a film of sodium in a photo- 
electric cell. I rely on the classical theory to conduct 
the light through the lenses and focus it in the cell; then 
I switch on to the quantum theory to make the light 
fetch out electrons from the sodium film to be collected 
in an electrometer. If I happen to transpose the two 
theories, the quantum theory convinces me that the light 
will never get concentrated in the cell and the classical 
theory shows that it is powerless to extract the elec- 
trons if it does get in. I have no logical reason for 
not using the theories this way round; only experience 
teaches me that I must not. Sir William Bragg was not 
overstating the case when he said that we use the classi- 
cal theory on Mondays, Wednesday and Fridays, and 
the quantum theory on Tuesdays, Thursdays and Satur- 
days. Perhaps that ought to make us feel a little sym- 
pathetic towards the man whose philosophy of the uni- 
verse takes one form on weekdays and another form on 
Sundays. 

In the last century — and I think also in this — there 
must have been many scientific men who kept their 
science and religion in watertight compartments. One 
set of beliefs held good in the laboratory and another set 
of beliefs in church, and no serious effort was made to 
harmonise them. The attitude is defensible. To discuss 
the compatibility of the beliefs would lead the scientist 
into regions of thought in which he was inexpert; and 
any answer he might reach would be undeserving of 



CLASSICAL AND QUANTUM LAWS 195 

strong confidence. Better admit that there was some 
truth both in science and religion; and if they must fight, 
let it be elsewhere than in the brain of a hard-working 
scientist. If we have ever scorned this attitude, Nemesis 
has overtaken us. For ten years we have had to divide 
modern science into two compartments; we have one set 
of beliefs in the classical compartment and another set 
of beliefs in the quantum compartment. Unfortunately 
our compartments are not watertight. 

We must, of course, look forward to an ultimate 
reconstruction of our conceptions of the physical world 
which will embrace both the classical laws and the 
quantum laws in harmonious association. There are still 
some who think that the reconciliation will be effected 
by a development of classical conceptions. But the 
physicists of what I may call "the Copenhagen school" 
believe that the reconstruction has to start at the other 
end, and that in the quantum phenomena we are getting 
down to a more intimate contact with Nature's way of 
working than in the coarse-grained experience which 
has furnished the classical laws. The classical school 
having become convinced of the existence of these uni- 
form lumps of action, speculates on the manufacture of 
the chopper necessary to carve off uniform lumps; the 
Copenhagen school on the other hand sees in these 
phenomena the insubstantial pageant of space, time and 
matter crumbling into grains of action. I do not think 
that the Copenhagen school has been mainly influenced 
by the immense difficulty of constructing a satisfactory 
chopper out of classical material; its view arises espe- 
cially from a study of the meeting point of quantum and 
classical laws. 

The classical laws are the limit to which the quantum 
laws tend when states of very high quantum number are 
concerned. 



i 9 6 THE QUANTUM THEORY 

This is the famous Correspondence Principle enun- 
ciated by Bohr. It was at first a conjecture based on 
rather slight hints; but as our knowledge of quantum 
laws has grown, it has been found that when we apply 
them to states of very high quantum number they con- 
verge to the classical laws, and predict just what the 
classical laws would predict. 

For an example, take a hydrogen atom with its elec- 
tron in a circular orbit of very high quantum number, 
that is to say far away from the proton. On Monday, 
Wednesday and Friday it is governed by classical laws. 
These say that it must emit a feeble radiation continu- 
ously, of strength determined by the acceleration it is 
undergoing and of period agreeing with its own period 
of revolution. Owing to the gradual loss of energy it 
will spiral down towards the proton. On Tuesday, 
Thursday and Saturday it is governed by quantum laws 
and jumps from one orbit to another. There is a 
quantum law that I have not mentioned which prescribes 
that (for circular orbits only) the jump must always be 
to the circular orbit next lower, so that the electron 
comes steadily down the series of steps without skipping 
any. Another law prescribes the average time between 
each jump and therefore the average time between the 
successive emissions of light. The small lumps of 
energy cast away at each step form light-waves of period 
determined by the h rule. 

"Preposterous! You cannot seriously mean that the 
electron does different things on different days of the 
week!" 

But did I say that it does different things? I used 
different words to describe its doings. I run down the 
stairs on Tuesday and slide down the banisters on 
Wednesday; but if the staircase consists of innumerable 



CLASSICAL AND QUANTUM LAWS 197 

infinitesimal steps, there is no essential difference in 
my mode of progress on the two days. And so it makes 
no difference whether the electron steps from one orbit 
to the next lower or comes down in a spiral when the 
number of steps is innumerably great. The succession 
of lumps of energy cast overboard merges into a con- 
tinuous outflow. If you had the formulae before you, 
you would find that the period of the light and the 
strength of radiation are the same whether calculated by 
the Monday or the Tuesday method — but only when 
the quantum number is infinitely great. The disagree- 
ment is not very serious when the number is moderately 
large; but for small quantum numbers the atom cannot 
sit on the fence. It has to decide between Monday 
(classical) and Tuesday (quantum) rules. It chooses 
Tuesday rules. 

If, as we believe, this example is typical, it indicates 
one direction which the reconstruction of ideas must 
take. We must not try to build up from classical con- 
ceptions, because the classical laws only become true and 
the conceptions concerned in them only become defined 
in the limiting case when the quantum numbers of the 
system are very large. We must start from new con- 
ceptions appropriate to low as well as to high numbered 
states; out of these the classical conceptions should 
emerge, first indistinctly, then definitely, as the number 
of the state increases, and the classical laws become 
more and more nearly true. " I cannot foretell the result 
of this remodelling, but presumably room must be 
found for a conception of "states", the unity of a 
state replacing the kind of tie expressed by classical 
forces. For low numbered states the current vocabulary 
of physics is inappropriate; at the moment we can 
scarcely avoid using it, but the present contradictoriness 



1 98 THE QUANTUM THEORY 

of our theories arises from this misuse. For such states 
space and time do not exist — at least I can see no reason 
to believe that they do. But it must be supposed that 
when high numbered states are considered there will 
be found in the new scheme approximate counterparts 
of the space and time of current conception — some- 
thing ready to merge into space and time when the 
state numbers are infinite. And simultaneously the inter- 
actions described by transitions of states will merge 
into classical forces exerted across space and time. So 
that in the limit the classical description becomes an 
available alternative. Now in practical experience we 
have generally had to deal with systems whose ties are 
comparatively loose and correspond to very high quan- 
tum numbers; consequently our first survey of the 
world has stumbled across the classical laws and our 
present conceptions of the world consist of those enti- 
ties which only take definite shape for high quantum 
numbers. But in the interior of the atom and molecule, 
in the phenomena of radiation, and probably also in the 
constitution of very dense stars such as the Companion 
of Sirius, the state numbers are not high enough to 
admit this treatment. These phenomena are now forcing 
us back to the more fundamental conceptions out of 
which the classical conceptions (sufficient for the other 
types of phenomena) ought to emerge as one extreme 
limit. 

For an example I will borrow a quantum conception 
from the next chapter. It may not be destined to sur- 
vive in the present rapid evolution of ideas, but at any 
rate it will illustrate my point. In Bohr's semi-classical 
model of the hydrogen atom there is an electron de- 
scribing a circular or elliptic orbit. This is only a model; 
the real atom contains nothing of the sort. The real 



CLASSICAL AND QUANTUM LAWS 199 

atom contains something which it has not entered into 
the mind of man to conceive, which has, however, been 
described symbolically by Schrodinger. This "some- 
thing" is spread about in a manner by no means com- 
parable to an electron describing an orbit. Now excite 
the atom into successively higher and higher quantum 
states. In the Bohr model the electron leaps into higher 
and higher orbits. In the real atom Schrodinger's 
"something" begins to draw itself more and more 
together until it begins sketchily to outline the Bohr 
orbit and even imitates a condensation running round. 
Go on to still higher quantum numbers, and Schro- 
dinger's symbol now represents a compact body moving 
round in the same orbit and the same period as the 
electron in Bohr's model, and moreover radiating 
according to the classical laws of an electron. And so 
when the quantum number reaches infinity, and the 
atom bursts, a genuine classical electron flies out. The 
electron, as it leaves the atom, crystallises out of Schro- 
dinger's mist like a genie emerging from his bottle. 



Chapter X 

THE NEW QUANTUM THEORY 

The conflict between quantum theory and classical 
theory becomes especially acute in the problem of the 
propagation of light. Here in effect it becomes a con- 
flict between the corpuscular theory of light and the 
wave theory. 

In the early days it was often asked, How large is a 
quantum of light? One answer is obtained by examining 
a star image formed with the great ioo-inch reflector 
at Mt. Wilson. The diffraction pattern shows that each 
emission from each atom must be filling the whole mir- 
ror. For if one atom illuminates one part only and 
another atom another part only, we ought to get the 
same effect by illuminating different parts of the mirror 
by different stars (since there is no particular virtue in 
using atoms from the same star) ; actually the diffraction 
pattern then obtained is not the same. The quantum 
must be large enough to cover a ioo-inch mirror. 

But if this same star-light without any artificial con- 
centration falls on a film of potassium, electrons will 
fly out each with the whole energy of a quantum. This 
is not a trigger action releasing energy already stored in 
the atom, because the amount of energy is fixed by the 
nature of thi light, not by the nature of the atom. A 
whole quantum of light energy must have gone into the 
atom and blasted away the electron. The quantum 
must be small enough to enter an atom. 

I do not think there is much doubt as to the ultimate 
origin of this contradiction. We must not think about 
space and time in connection with an individual quan- 

200 



WAVE-THEORY OF MATTER 201 

turn; and the extension of a quantum in space has no 
real meaning. To apply these conceptions to a single 
quantum is like reading the Riot Act to one man. A 
single quantum has not travelled 50 billion miles from 
Sirius; it has not been 8 years on the way. But when 
enough quanta are gathered to form a quorum there 
will be found among them statistical properties which 
are the genesis of the 50 billion miles' distance of Sirius 
and the 8 years' journey of the light. 

Wave-Theory of Matter. It is comparatively easy to 
realise what we have got to do. It is much more diffi- 
cult to start to do it. Before we review the attempts 
in the last year or two to grapple with this problem we 
shall briefly consider a less drastic method of progress 
initiated by De Broglie. For the moment we shall be 
content to accept the mystery as a mystery. Light, we 
will say, is an entity with the wave property of spread- 
ing out to fill the largest object glass and with all the 
well-known properties of diffraction and interference; 
simultaneously it is an entity with the corpuscular or 
bullet property of expending its whole energy on one 
very small target. We can scarcely describe such an 
entity as a wave or as a particle; perhaps as a com- 
promise we had better call it a "wavicle". 

There is nothing new under the sun, and this latest. 
volte-face almost brings us back to Newton's theory of 
light — a curious mixture of corpuscular and wave-theory. 
There is perhaps a pleasing sentiment in this "return 
to Newton". But to suppose that Newton's scientific 
reputation is especially vindicated by De Broglie's 
theory of light, is as absurd as to suppose that it is 
shattered by Einstein's theory of gravitation. There 
was no phenomenon known to Newton which could not 



202 THE NEW QUANTUM THEORY 

be amply covered by the wave-theory; and the clearing 
away of false evidence for a partly corpuscular theory, 
which influenced Newton, is as much a part of scientific 
progress as the bringing forward of the (possibly) true 
evidence, which influences us to-day. To imagine that 
Newton's great scientific reputation is tossing up and 
down in these latter-day revolutions is to confuse science 
with omniscience. 

To return to the wavicle. — If that which we have 
commonly regarded as a wave partakes also of the 
nature of a particle, may not that which we have com- 
monly regarded as a particle partake also of the nature 
of a wave? It was not until the present century that 
experiments were tried of a kind suitable to bring out 
the corpuscular aspect of the nature of light; perhaps 
experiments may still be possible which will bring out 
a wave aspect of the nature of an electron. 

So, as a first step, instead of trying to clear up the 
mystery we try to extend it. Instead of explaining how 
anything can possess simultaneously the incongruous 
properties of wave and particle we seek to show experi- 
mentally that these properties are universally associated. 
There are no pure waves and no pure particles. 

The characteristic of a wave-theory is the spreading 
of a ray of light after passing through a narrow aper- 
ture — a well-known phenomenon called diffraction. The 
scale of the phenomenon is proportional to the wave- 
length of the light. De Broglie has shown us how to 
calculate the lengths of the waves (if any) associated 
with an electron, i.e. considering it to be no longer a pure 
particle but a wavicle. It appears that in some circum- 
stances the scale of the corresponding diffraction effects 
will not be too small for experimental detection. There 
are now a number of experimental results quoted as 



WAVE-THEORY OF MATTER 203 

verifying this prediction. I scarcely know whether they 
are yet to be considered conclusive, but there does seem 
to be serious evidence that in the scattering of electrons 
by atoms phenomena occur which would not be pro- 
duced according to the usual theory that electrons are 
purely corpuscular. These effects analogous to the 
diffraction and interference of light carry us into the 
stronghold of the wave-theory. Long ago such phe- 
nomena ruled out all purely corpuscular theories of 
light; perhaps to-day we are finding similar phenomena 
which will rule out all purely corpuscular theories of 
matter.* 

A similar idea was entertained in a "new statistical 
mechanics" developed by Einstein and Bose — at least 
that seems to be the physical interpretation of the highly 
abstract mathematics of their theory. As so often hap- 
pens the change from the classical mechanics, though 
far-reaching in principle, gave only insignificant cor- 
rections when applied to ordinary practical problems. 
Significant differences could only be expected in matter 
much denser than anything yet discovered or imagined. 
Strange to say, just about the time when it was realised 
that very dense matter might have strange properties 
different from those expected according to classical 
conceptions, very dense matter was found in the uni- 
verse. Astronomical evidence seems to leave practically 
no doubt that in the so-called white dwarf stars the 
density of matter far transcends anything of which we 
have terrestrial experience; in the Companion of Sirius, 
for example, the density is about a ton to the cubic inch. 
This condition is explained by the fact that the high 
temperature and correspondingly intense agitation of 

*The evidence is much stronger now than when the lectures were 
delivered. 



204 THE NEW QUANTUM THEORY 

the material breaks up (ionises) the outer electron sys- 
tems of the atoms, so that the fragments can be packed 
much more closely together. At ordinary temperatures 
the minute nucleus of the atom is guarded by outposts 
of sentinel electrons which ward off other atoms from 
close approach even under the highest pressures; but at 
stellar temperatures the agitation is so great that the 
electrons leave their posts and run all over the place. 
Exceedingly tight packing then becomes possible under 
high enough pressure. R. H. Fowler has found that in 
the white dwarf stars the density is so great that classi- 
cal methods are inadequate and the new statistical 
mechanics must be used. In particular he has in this 
way relieved an anxiety which had been felt as to their 
ultimate fate; under classical laws they seemed to be 
heading towards an intolerable situation — the star could 
not stop losing heat, but it would have insufficient energy 
to be able to cool down!* 

Transition to a New Theory. By 1925 the machinery 
of current theory had developed another flaw and was 
urgently calling for reconstruction; Bohr's model of the 
atom had quite definitely broken down. This is the 
model, now very familiar, which pictures the atom as 
a kind of solar system with a central positively charged 
nucleus and a number of elecrons describing orbits about 
it like planets, the important feature being that the 
possible orbits are limited by the rules referred to on 
p. 190. Since each line in the spectrum of the atom is 
emitted by the jump of an electron between two par- 

* The energy is required because on cooling down the matter must 
regain a more normal density and this involves a great expansion of 
volume of the star. In the expansion work has to be done against the 
force of gravity. 



TRANSITION TO A NEW THEORY 205 

ticular orbits, the classification of the spectral lines must 
run parallel with the classification of the orbits by their 
quantum numbers in the model. When the spectro- 
scopists started to unravel the various series of lines in 
the spectra they found it possible to assign an orbit 
jump for every line — they could say what each line 
meant in terms of the model. But now questions of 
finer detail have arisen for which this correspondence 
ceases to hold. One must not expect too much from a 
model, and it would have been no surprise if the model 
had failed to exhibit minor phenomena or if its accuracy 
had proved imperfect. But the kind of trouble now 
arising was that only two orbit jumps were provided 
in the model to represent three obviously associated 
spectral lines; and so on. The model which had been 
so helpful in the interpretation of spectra up to a point, 
suddenly became altogether misleading; and spectro- 
scopists were forced to turn away from the model and 
complete their classification of lines in a way which 
ignored it. They continued to speak of orbits and 
orbit jumps but there was no longer a complete one- 
to-one correspondence with the orbits shown in the 
model.* 

The time was evidently ripe for the birth of a new 
theory. The situation then prevailing may be summar- 
ised as follows : 

(1) The general working rule was to employ the 
classical laws with the supplementary proviso that 
whenever anything of the nature of action appears it 

*Each orbit or state of the atom requires three (or, for later refine- 
ments, four) quantum numbers to define it. The first two quantum 
numbers are correctly represented in the Bohr model ; but the third 
number which discriminates the different lines forming a doublet or 
multiplet spectrum is represented wrongly — a much more serious failure 
than if it were not represented at all. 



206 THE NEW QUANTUM THEORY 

must be made equal to h ) or sometimes to an integral mul- 
tiple of h. 

(2) The proviso often led to a self-contradictory use 
of the classical theory. Thus in the Bohr atom the 
acceleration of the electron in its orbit would be gov- 
erned by classical electrodynamics whilst its radiation 
would be governed by the h rule. But in classical elec- 
trodynamics the acceleration and the radiation are indis- 
solubly connected. 

(3) The proper sphere of classical laws was known. 
They are a form taken by the more general laws in a 
limiting case, viz. when the number of quanta concerned 
is very large. Progress in the investigation of the com- 
plete system of more general laws must not be ham- 
pered by classical conceptions which contemplate only the 
limiting case. 

(4) The present compromise involved the recognition 
that light has both corpuscular and wave properties. 
The same idea seems to have been successfully extended 
to matter and confirmed by experiment. But this success 
only renders the more urgent some less contradictory 
way of conceiving these properties. 

(5) Although the above working rule had generally 
been successful in its predictions, it was found to give 
a distribution of electron orbits in the atom differing in 
some essential respects from that deduced spectroscopi- 
cally. Thus a reconstruction was required not only to 
remove logical objections but to meet the urgent de- 
mands of practical physics. 

Development of the New Quantum Theory. The "New 
Quantum Theory" originated in a remarkable paper 
by Heisenberg in the autumn of 1925. I am writing 
the first draft of this lecture just twelve months after 



DEVELOPMENT OF NEW THEORY 207 

the appearance of the paper. That does not give long 
for development; nevertheless the theory has already 
gone through three distinct phases associated with the 
names of Born and Jordan, Dirac, Schrodinger. My 
chief anxiety at the moment is lest another phase of 
reinterpretation should be reached before the lecture 
can be delivered. In an ordinary way we should describe 
the three phases as three distinct theories. The pioneer 
work of Heisenberg governs the whole, but the three 
theories show wide differences of thought. The first 
entered on 'the new road in a rather matter-of-fact 
way; the second was highly transcendental, almost 
mystical; the third seemed at first to contain a reac- 
tion towards classical ideas, but that was probably a 
false impression. You will realise the anarchy of 
this branch of physics when three successive pre- 
tenders seize the throne in twelve months; but you 
will not realise the steady progress made in that time 
unless you turn to the mathematics of the subject. 
As regards philosophical ideas the three theories are 
poles apart; as regards mathematical content they are 
one and the same. Unfortunately the mathematical 
content is just what I am forbidden to treat of in these 
lectures. 

I am, however, going to transgress to the extent of 
writing down one mathematical formula for you to con- 
template; I shall not be so unreasonable as to expect 
you to understand it. All authorities seem to be agreed 
that at, or nearly at, the root of everything in the phy- 
sical world lies the mystic formula 

qp—pq = ih/2Tz 

We do not yet understand that; probably if we could 
understand it we should not think it so fundamental. 



208 THE NEW QUANTUM THEORY 

Where the trained mathematician has the advantage is 
that he can use it, and in the past year or two it has 
been used in physics with very great advantage indeed. 
It leads not only to those phenomena described by the 
older quantum laws such as the h rule, but to many 
related phenomena which the older formulation could 
not treat. 

On the right-hand side, besides h (the atom of action) 
and the merely numerical factor 2tt, there appears i (the 
square root of — i) which may seem rather mystical. 
But this is only a well-known subterfuge; and far back 
in the last century physicists and engineers were well 
aware that V — i in their formulae was a kind of sig- 
nal to look out for waves or oscillations. The right- 
hand side contains nothing unusual, but the left-hand side 
baffles imagination. We call q and p co-ordinates and mo- 
menta, borrowing our vocabulary from the world of 
space and time and other coarse-grained experience; 
but that gives no real light on their nature, nor does 
it explain why qp is so ill-behaved as to be unequal 
to pq. 

It is here that the three theories differ most essen- 
tially. Obviously q and p cannot represent simple 
numerical measures, for then qp — pq would be zero. 
For Schrodinger p is an operator. His "momentum" 
is not a quantity but a signal to us to perform a certain 
mathematical operation on any quantities which may 
follow. For Born and Jordan p is a matrix — not one 
quantity, nor several quantities, but an infinite number 
of quantities arranged in systematic array. For Dirac 
p is a symbol without any kind of numerical interpreta- 
tion; he calls it a ^-number, which is a way of saying 
that it is not a number at all. 

I venture to think that there is an idea implied in 



DEVELOPMENT OF NEW THEORY 209 

Dirac^s treatment which may have great philosophical 
significance, independently of any question of success in 
this particular application. The idea is that in digging 
deeper and deeper into that which lies at the base of 
physical phenomena we must be prepared to come to 
entities which, like many things in our conscious experi- 
ence, are not measurable by numbers in any way; and 
further it suggests how exact science, that is to say the 
science of phenomena correlated to measure-numbers, 
can be founded on such a basis. 

One of the greatest changes in physics between the 
nineteenth century and the present day has been the 
change in our ideal of scientific explanation. It was the 
boast of the Victorian physicist that he would not claim 
to understand a thing until he could make a model of 
it; and by a model he meant something constructed of 
levers, geared wheels, squirts, or other appliances 
familiar to an engineer. Nature in building the universe 
was supposed to be dependent on just the same kind of 
resources as any human mechanic; and when the physi- 
cist sought an explanation of phenomena his ear was 
straining to catch the hum of machinery. The man who 
could make gravitation out of cog-wheels would have 
been a hero in the Victorian age. 

Nowadays we do not encourage the engineer to build 
the world for us out of his material, but we turn to the 
mathematician to build it out of his material. Doubtless 
the mathematician is a loftier being than the engineer, 
but perhaps even he ought not to be entrusted with the 
Creation unreservedly. We are dealing in physics with 
a symbolic world, and we can scarcely avoid employing 
the mathematician who is the professional wielder of 
symbols; but he must rise to the full opportunities of the 
responsible task entrusted to him and not indulge too 



210 THE NEW QUANTUM THEORY 

freely his own bias for symbols with an arithmetical 
interpretation. If we are to discern controlling laws of 
Nature not dictated by the mind it would seem neces- 
sary to escape as far as possible from the cut-and-dried 
framework into which the mind is so ready to force 
everything that it experiences. 

I think that in principle Dirac's method asserts this 
kind of emancipation. He starts with basal entities 
inexpressible by numbers or number-systems and his 
basal laws are symbolic expressions unconnected with 
arithmetical operations. The fascinating point is that 
as the development proceeds actual numbers are exuded 
from the symbols. Thus although p and q individually 
have no arithmetical interpretation, the combination 
qp — pq has the arithmetical interpretation expressed by 
the formula above quoted. By furnishing numbers, 
though itself non-numerical, such a theory can well be 
the basis for the measure-numbers studied in exact 
science. The measure-numbers, which are all that we 
glean from a physical survey of the world, cannot be 
the whole world; they may not even be so much of it 
as to constitute a self-governing unit. This seems the 
natural interpretation of Dirac's procedure in seeking 
the governing laws of exact science in a non-arithmetical 
calculus. 

I am afraid it is a long shot to predict anything like 
this emerging from Dirac's beginning; and for the 
moment Schrodinger has rent much of the mystery from 
the />'s and qs by showing that a less transcendental 
interpretation is adequate for present applications. But 
I like to think that we may have not yet heard the last 
of the idea. 

Schrodinger's theory is now enjoying the full tide 
of popularity, partly because of intrinsic merit, but also, 



OUTLINE OF SCHRODINGER'S THEORY 211 

I suspect, partly because it is the only one of the three 
that is simple enough to be misunderstood. Rather 
against my better judgment I will try to give a rough 
impression of the theory. It would probably be wiser 
to nail up over the door of the new quantum theory a 
notice, "Structural alterations in progress — No admit- 
tance except on business", and particularly to warn the 
doorkeeper to keep out prying philosophers. I will, 
however, content myself with the protest that, whilst 
Schrodinger's theory is guiding us to sound and rapid 
progress in many of the mathematical problems con- 
fronting us and is indispensable in its practical utility, 
I do not see the least likelihood that his ideas will sur- 
vive long in their present form. 

Outline of Schrodinger's Theory. Imagine a sub-aether 
whose surface is covered with ripples. The oscillations 
of the ripples are a million times faster than those of 
visible light — too fast to come within the scope of our 
gross experience. Individual ripples are beyond our 
ken; what we can appreciate is a combined effect — when 
by convergence and coalescence the waves conspire to 
create a disturbed area of extent large compared with 
individual ripples but small from our own Brobding- 
nagian point of view. Such a disturbed area is recog- 
nised as a material particle; in particular it can be an 
electron. 

The sub-aether is a dispersive medium, that is to say 
the ripples do not all travel with the same velocity; like 
water-ripples their speed depends on their wave-length 
or period. Those of shorter period travel faster. More- 
over the speed may be modified by local conditions. 
This modification is the counterpart in Schrodinger's 
theory of a field of force in classical physics. It will 



212 THE NEW QUANTUM THEORY 

readily be understood that if we are to reduce all phe- 
nomena to a propagation of waves, then the influence 
of a body on phenomena in its neighbourhood (com- 
monly described as the field of force caused by its 
presence) must consist in a modification of the propa- 
gation of waves in the region surrounding it. 

We have to connect these phenomena in the sub- 
aether with phenomena in the plane of our gross ex- 
perience. As already stated, a local stormy region is 
detected by us as a particle; to this we now add that the 
frequency (number of oscillations per second) of the 
waves constituting the disturbance is recognised by us 
as the energy of the particle. We shall presently try to 
explain how the period manages to manifest itself to us 
in this curiously camouflaged way; but however it comes 
about, the recognition of a frequency in the sub-aether 
as an energy in gross experience gives at once the con- 
stant relation between period and energy which we have 
called the h rule. 

Generally the oscillations in the sub-aether are too 
rapid for us to detect directly; their frequency reaches 
the plane of ordinary experience by affecting the speed 
of propagation, because the speed depends (as already 
stated) on the wave-length or frequency. Calling the 
frequency v, the equation expressing the law of propa- 
gation of the ripples will contain a term in v. There will 
be another term expressing the modification caused by 
the "field of force" emanating from the bodies present 
in the neighbourhood. This can be treated as a kind of 
spurious v, since it emerges into our gross experience 
by the same method that v does. If v produces those 
phenomena which make us recognise it as energy, the 
spurious v will produce similar phenomena correspond- 
ing to a spurious kind of energy. Clearly the latter will 



OUTLINE OF SCHRODINGER'S THEORY 213 

be what we call potential energy, since it originates from 
influences attributable to the presence of surrounding 
objects. 

Assuming that we know both the real v and the 
spurious or potential v for our ripples, the equation of 
wave-propagation is settled, and we can proceed to solve 
any problem concerning wave-propagation. In particular 
we can solve the problem as to how the stormy areas 
move about. This gives a remarkable result which 
provides the first check on our theory. The stormy 
areas (if small enough) move under precisely the same 
laws that govern the motions of particles in classical 
mechanics. The equations for the motion of a wave- 
group with given frequency and potential frequency are 
the same as the classical equations of motion of a par- 
ticle with the corresponding energy and potential energy. 

It has to be noticed that the velocity of a stormy area 
or group of waves is not the same as the velocity of an 
individual wave. This is well known in the study of 
water-waves as the distinction between group-velocity 
and wave-velocity. It is the group-velocity that is ob- 
served by us as the motion of the material particle. 

We should have gained very little if our theory did 
no more than re-establish the results of classical me- 
chanics on this rather fantastic basis. Its distinctive 
merits begin to be apparent when we deal with pheno- 
mena not covered by classical mechanics. We have 
considered a stormy area of so small extent that its 
position is as definite as that of a classical particle, but 
we may also consider an area of wider extent. No 
precise delimitation can be drawn between a large area 
and a small area, so that we shall continue to associate 
the idea of a particle with it; but whereas a small 
concentrated storm fixes the position of the particle 



214 THE NEW QUANTUM THEORY 

closely, a more extended storm leaves it very vague. If 
we try to interpret an extended wave-group in classical 
language we say that it is a particle which is not at any 
definite point of space, but is loosely associated with a 
wide region. 

Perhaps you may think that an extended stormy area 
ought to represent diffused matter in contrast to a con- 
centrated particle. That is not Schrodinger's theory. 
The spreading is not a spreading of density; it is an 
indeterminacy of position, or a wider distribution of the 
probability that the particle lies within particular limits 
of position. Thus if we come across Schrodinger waves 
uniformly filling a vessel, the interpretation is not that 
the vessel is filled with matter of uniform density, but 
that it contains one particle which is equally likely to be 
anywhere. 

The first great success of this theory was in repre- 
senting the emission of light from a hydrogen atom — 
a problem far outside the scope of classical theory. The 
hydrogen atom consists of a proton and electron which 
must be translated into their counterparts in the sub- 
aether. We are not interested in what the proton is 
doing, so we do not trouble about its representation by 
waves; what we want from it is its field of force, that is 
to say, the spurious v which it provides in the equation 
of wave-propagation for the electron. The waves 
travelling in accordance with this equation constitute 
Schrodinger's equivalent for the electron; and any solu- 
tion of the equation will correspond to some possible 
state of the hydrogen atom. Now it turns out that 
(paying attention to the obvious physical limitation that 
the waves must not anywhere be of infinite amplitude) 
solutions of this wave-equation only exist for waves with 
particular frequencies. Thus in a hydrogen atom the 



OUTLINE OF SCHRODINGER'S THEORY 215 

sub-aethereal waves are limited to a particular discrete 
series of frequencies. Remembering that a frequency 
in the sub-aether means an energy in gross experience, 
the atom will accordingly have a discrete series of pos- 
sible energies. It is found that this series of energies 
is precisely the same as that assigned by Bohr from his 
rules of quantisation (p. 191). It is a considerable 
advance to have determined Jiese energies by a wave- 
theory instead of by an inexplicable mathematical rule. 
Further, when applied to more complex atoms Schro- 
dinger's theory succeeds on those points where the Bohr 
model breaks down; it always gives the right number of 
energies or "orbits" to provide one orbit jump for each 
observed spectral line. 

It is, however, an advantage not to pass from wave- 
frequency to classical energy at this stage, but to follow 
the course of events in the sub-aether a little farther. 
It would be difficult to think of the electron as having 
two energies (i.e. being in two Bohr orbits) simultane- 
ously; but there is nothing to prevent waves of two dif- 
ferent frequencies being simultaneously present in the 
sub-aether. Thus the wave-theory allows us easily to 
picture a condition which the classical theory could only 
describe in paradoxical terms. Suppose that two sets 
of waves are present. If the difference of frequency is 
not very great the two systems of waves will produce 
"beats". If two broadcasting stations are transmitting 
on wave-lengths near together we hear a musical note 
or shriek resulting from the beats of the two carrier 
waves; the individual oscillations are too rapid to affect 
the ear, but they combine to give beats which are slow 
enough to affect the ear. In the same way the individual 
wave-systems in the sub-aether are composed of oscilla- 
tions too rapid to affect our gross senses ; but their beats 



216 THE NEW QUANTUM THEORY 

are sometimes slow enough to come within the octave 
covered by the eye. These beats are the source of the 
light coming from the hydrogen atom, and mathematical 
calculation shows that their frequencies are precisely 
those of the observed light from hydrogen. Hetero- 
dyning of the radio carrier waves produces sound; 
heterodyning of the sub-aethereal waves produces light. 
Not only does this theory give the periods of the dif- 
ferent lines in the spectra, but it also predicts their in- 
tensities — a problem which the older quantum theory had 
no means of tackling. It should, however, be under- 
stood that the beats are not themselves to be identified 
with light-waves; they are in the sub-aether, whereas 
light-waves are in the aether. They provide the oscil- 
lating source which in some way not yet traced sends out 
light-waves of its own period. 

What precisely is the entity which we suppose to be 
oscillating when we speak of the waves in the sub- 
aether? It is denoted by op, and properly speaking we 
should regard it as an elementary indefinable of the 
wave-theory. But can we give it a classical interpreta- 
tion of any kind? It seems possible to interpret it as a 
probability. The probability of the particle or electron 
being within a given region is proportional to the amount 
of ip in that region. So that if ip is mainly concentrated 
in one small stormy area, it is practically certain that 
the electron is there; we are then able to localise it 
definitely and conceive of it as a classical particle. But 
the ip-waves of the hydrogen atom are spread about 
all over the atom; and there is no definite localisation of 
the electron, though some places are more probable than 
others.* 

* The probability is often stated to be proportional to ty 2 , instead of 
\p, as assumed above. The whole interpretation is very obscure, but it 



OUTLINE OF SCHRODINGER'S THEORY 217 

Attention must be called to one highly important 
consequence of this theory. A small enough stormy 
area corresponds very nearly to a particle moving about 
under the classical laws of motion; it would seem there- 
fore that a particle definitely localised as a moving point 
is stricdy the limit when the stormy area is reduced to 
a point. But curiously enough by continually reducing 
the area of the storm we never quite reach the ideal 
classical particle; we approach it and then recede from 
it again. We have seen that the wave-group moves like 
a particle (localised somewhere within the area of the 
storm) having an energy corresponding to the frequency 
of the waves; therefore to imitate a particle exactly, not 
only must the area be reduced to a point but the group 
must consist of waves of only one frequency. The two 
conditions are irreconcilable. With one frequency we 
can only have an infinite succession of waves not ter- 
minated by any boundary. A boundary to the group is 
provided by interference of waves of slightly different 
length, so that while reinforcing one another at the 
centre they cancel one another at the boundary. Roughly 
speaking, if the group has a diameter of 1000 wave- 
lengths there must be a range of wave-length of o-i per 
cent., so that 1000 of the longest waves and 1001 of 
the shortest occupy the same distance. If we take a 
more concentrated stormy area of diameter 10 wave- 

seems to depend on whether you are considering the probability after 
you know what has happened or the probability for the purposes of 
prediction. The ijj 2 is obtained by introducing two symmetrical systems 
of ij>-waves travelling in opposite directions in time; one of these must 
presumably correspond to probable inference from what is known (or 
is stated) to have been the condition at a later time. Probability neces- 
sarily means "probability in the light of certain given information", so 
that the probability cannot possibly be represented by the same function 
in different classes of problems with different initial data. 



2i 8 THE NEW QUANTUM THEORY 

lengths the range is increased to 10 per cent.; 10 of 
the longest and 1 1 of the shortest waves must extend the 
same distance. In seeking to make the position of the 
particle more definite by reducing the area we make its 
energy more vague by dispersing the frequencies of the 
waves. So our particle can never have simultaneously 
a perfectly definite position and a perfectly definite 
energy; it always has a vagueness of one kind or the 
other unbefitting a classical particle. Hence in delicate 
experiments we must not under any circumstances expect 
to find particles behaving exactly as a classical particle 
was supposed to do — a conclusion which seems to be in 
accordance with the modern experiments on diffraction 
of electrons already mentioned. 

We remarked that Schrodinger's picture of the hy- 
drogen atom enabled it to possess something that would 
be impossible on Bohr's theory, viz. two energies at 
once. For a particle or electron this is not merely per- 
missive, but compulsory — otherwise we can put no limits 
to the region where it may be. You are not asked to 
imagine the state of a particle with several energies; 
what is meant is that our current picture of an electron 
as a particle with single energy has broken down, and 
we must dive below into the sub-aether if we wish to 
follow the course of events. The picture of a particle 
may, however, be retained when we are not seeking high 
accuracy; if we do not need to know the energy more 
closely than I per cent., a series of energies ranging 
over i per cent, can be treated as one definite energy. 

Hitherto I have only considered the waves correspond- 
ing to one electron; now suppose that we have a prob- 
lem involving two electrons. How shall they be repre- 
sented? "Surely, that is simple enough! We have only 
to take two stormy areas instead of one." I am afraid 



OUTLINE OF SCHRODINGER'S THEORY 219 

not. Two stormy areas would correspond to a single 
electron uncertain as to which area it was located in. 
So long as there is the faintest probability of the first 
electron being in any region, we cannot make the Schro- 
dinger waves there represent a probability belonging to 
a second electron. Each electron wants the whole of 
three-dimensional space for its waves; so Schrodinger 
generously allows three dimensions for each of them. 
For two electrons he requires a six-dimensional sub- 
aether. He then successfully applies his method on the 
same lines as before. I think you will see now that 
Schrodinger has given us what seemed to be a com- 
prehensible physical picture only to snatch it away again. 
His sub-aether does not exist in physical space; it is in 
a "configuration space" imagined by the mathematician 
for the purpose of solving his problems, and imagined 
afresh with different numbers of dimensions according 
to the problem proposed. It was only an accident 
that in the earliest problems considered the configu- 
ration space had a close correspondence with physical 
space, suggesting some degree of objective reality 
of the waves. Schrodinger's wave-mechanics is not 
a physical theory but a dodge — and a very good dodge 
too. 

The fact is that the almost universal applicability of 
this wave-mechanics spoils all chance of our taking it 
seriously as a physical theory. A delightful illustration 
of this occurs incidentally in the work of Dirac. In one 
of the problems, which he solves by Schrodinger waves, 
the frequency of the waves represents the number of 
systems of a given kind. The wave-equation is formu- 
lated and solved, and (just as in the problem of the 
hydrogen atom) it is found that solutions only exist for 
a series of special values of the frequency. Consequently 



220 THE NEW QUANTUM THEORY 

the number of systems of the kind considered must 
have one of a discrete series of values. In Dirac's 
problem the series turns out to be the series of integers. 
Accordingly we infer that the number of systems must 
be either i, 2, 3, 4, . . ., but can never be 2% f° r 
example. It is satisfactory that the theory should give 
a result so well in accordance with our experience ! 
But we are not likely to be persuaded that the true 
explanation of why we count in integers is afforded by a 
system of waves. 

Principle of Indeterminacy. My apprehension lest a 
fourth version of the new quantum theory should 
appear before the lectures were delivered was not ful- 
filled; but a few months later the theory definitely 
entered on a new phase. It was Heisenberg again who 
set in motion the new development in the summer of 
1927, and the consequences were further elucidated by 
Bohr. The outcome of it is a fundamental general 
principle which seems to rank in importance with the 
principle of relativity. I shall here call it the "principle 
of indeterminacy". 

The gist of it can be stated as follows : a particle may 
have position or it may have velocity but it cannot in any 
exact sense have both. 

If we are content with a certain margin of inaccuracy 
and if we are content with statements that claim no 
certainty but only high probability, then it is possible 
to ascribe both position and velocity to a particle. But 
if we strive after a more accurate specification of position 
a very remarkable thing happens; the greater accuracy 
can be attained, but it is compensated by a greater 
inaccuracy in the specification of the velocity. Similarly 
if the specification of the velocity is made more accurate 
the position becomes less determinate. 



PRINCIPLE OF INDETERMINACY 221 

Suppose for example that we wish to know the posi- 
tion and velocity of an electron at a given moment. 
Theoretically it would be possible to fix the position with 
a probable error of about 1/1000 of a millimetre and 
the velocity with a probable error of 1 kilometre per 
second. But an error of 1/1000 of a millimetre is large 
compared with that of some of our space measurements; 
is there no conceivable way of fixing the position to 
1/10,000 of a millimetre? Certainly; but in that case it 
will only be possible to fix the velocity with an error of 
10 kilometres per second. 

The conditions of our exploration of the secrets of 
Nature are such that the more we bring to light the 
secret of position the more the secret of velocity is 
hidden. They are like the old man and woman in the 
weather-glass; as one comes out of one door, the other 
retires behind the other door. When we encounter un- 
expected obstacles in finding out something which we 
wish to know, there are two possible courses to take. It 
may be that the right course is to treat the obstacle 
as a spur to further efforts; but there is a second 
possibility — that we have been trying to find some- 
thing which does not exist. You will remember that 
that was how the relativity theory accounted for the 
apparent concealment of our velocity through the 
aether. 

When the concealment is found to be perfectly sys- 
tematic, then we must banish the corresponding entity 
from the physical world. There is really no option. 
The link with our consciousness is completely broken. 
When we cannot point to any causal effect on anything 
that comes into our experience, the entity merely becomes 
part of the unknown — undifferentiated from the rest of 
the vast unknown. From time to time physical discover- 
ies are made; and new entities, coming out of the un- 



222 THE NEW QUANTUM THEORY 

known, become connected to our experience and are duly 
named. But to leave a lot of unattached labels floating 
in the as yet undifferentiated unknown in the hope that 
they may come in useful later on, is no particular sign 
of prescience and is not helpful to science. From this 
point of view we assert that the description of the posi- 
tion and velocity of an electron beyond a limited num- 
ber of places of decimals is an attempt to describe some- 
thing that does not exist; although curiously enough the 
description of position or of velocity if it had stood alone 
might have been allowable. 

Ever since Einstein's theory showed the importance 
of securing that the physical quantities which we talk 
about are actually connected to our experience, we have 
been on our guard to some extent against meaningless 
terms. Thus distance is defined by certain operations of 
measurement and not with reference to nonsensical con- 
ceptions such as the "amount of emptiness" between 
two points. The minute distances referred to in atomic 
physics naturally aroused some suspicion, since it is not 
always easy to say how the postulated measurements 
could be imagined to be carried out. I would not like 
to assert that this point has been cleared up; but at any 
rate it did not seem possible to make a clean sweep of 
all minute distances, because cases could be cited in which 
there seemed no natural limit to the accuracy of deter- 
mination of position. Similarly there are ways of 
determining momentum apparently unlimited in accuracy. 
What escaped notice was that the two measurements 
interfere with one another in a systematic way, so that 
the combination of position with momentum, legitimate 
on the large scale, becomes indefinable on the small 
scale. The principle of indeterminacy is scientifically 
stated as follows: if q is a co-ordinate and p the corre- 



PRINCIPLE OF INDETERMINACY 223 

sponding momentum, the necessary uncertainty of our 
knowledge of q multiplied by the uncertainty of p is of 
the order of magnitude of the quantum constant h. 

A general kind of reason for this can be seen without 
much difficulty. Suppose it is a question of knowing 
the position and momentum of an electron. So long as 
the electron is not interacting with the rest of the uni- 
verse we cannot be aware of it. We must take our 
chance of obtaining knowledge of it at moments when it 
is interacting with something and thereby producing 
effects that can be observed. But in any such interaction 
a complete quantum is involved; and the passage of this 
quantum, altering to an important extent the conditions 
at the moment of our observation, makes the information 
out of date even as we obtain it. 

Suppose that (ideally) an electron is observed under 
a powerful microscope in order to determine its position 
with great accuracy. For it to be seen at all it must be 
illuminated and scatter light to reach the eye. The least 
it can scatter is one quantum. In scattering this it re- 
ceives from the light a kick of unpredictable amount; 
we can only state the respective probabilities of kicks 
of different amounts. Thus the condition of our ascer- 
taining the position is that we disturb the electron in an 
incalculable way which will prevent our subsequently as- 
certaining how much momentum it had. However, we 
shall be able to ascertain the momentum with an uncer- 
tainty represented by the kick, and if the probable kick 
is small the probable error will be small. To keep the 
kick small we must use a quantum of smali energy, that 
is to say, light of long wave-length. But to use long 
wave-length reduces the accuracy of our microscope. 
The longer the waves, the larger the diffraction images. 
And it must be remembered that it takes a great many 



224 THE NEW QUANTUM THEORY 

quanta to outline the diffraction image; our one scattered 
quantum can only stimulate one atom in the retina of 
the eye, at some haphazard point within the theoretical 
diffraction image. Thus there will be an uncertainty in 
our determination of position of the electron propor- 
tional to the size of the diffraction image. We are in a 
dilemma. We can improve the determination of the 
position with the microscope by using light of shorter 
wave-length, but that gives the electron a greater kick 
and spoils the subsequent determination of momentum. 

A picturesque illustration of the same dilemma is 
afforded if we imagine ourselves trying to see one of the 
electrons in an atom. For such finicking work it is no 
use employing ordinary light to see with; it is far too 
gross, its wave-length being greater than the whole 
atom. We must use fine-grained illumination and train 
our eyes to see with radiation of short wave-length — 
with X-rays in fact. It is well to remember that X-rays 
have a rather disastrous effect on atoms, so we had better 
use them sparingly. The least amount we can use is one 
quantum. Now, if we are ready, will you watch, whilst 
I flash one quantum of X-rays on to the atom? I may 
not hit the electron the first time; in that case, of course, 
you will not see it. Try again; this time my quantum 
has hit the electron. Look sharp, and notice where it is. 
Isn't it there? Bother! I must have blown the electron 
out of the atom. 

This is not a casual difficulty; it is a cunningly 
arranged plot — a plot to prevent you from seeing 
something that does not exist, viz. the locality of the 
electron within the atom. If I use longer waves which 
do no harm, they will not define the electron sharply 
enough for you to see where it is. In shortening the wave- 
length, just as the light becomes fine enough its quan- 



A NEW EPISTEMOLOGY 225 

turn becomes too rough and knocks the electron out of 
the atom. 

Other examples of the reciprocal uncertainty have 
been given, and there seems to be no doubt that it is 
entirely general. The suggestion is that an association 
of exact position with exact momentum can never be 
discovered by us because there is no such thing in Nature. 
This is not inconceivable. Schrodinger's model of the 
particle as a wave-group gives a good illustration of how 
it can happen. We have seen (p. 217) that as the posi- 
tion of a wave-group becomes more defined the energy 
(frequency) becomes more indeterminate, and vice versa. 
I think that that is the essential value of Schrodinger's 
theory; it refrains from attributing to a particle a kind 
of determinacy which does not correspond to anything 
in Nature. But I would not regard the principle of 
indeterminacy as a result to be deduced from Schro- 
dinger's theory; it is the other way about. The principle 
of indeterminacy, like the principle of relativity, repre- 
sents the abandonment of a mistaken assumption which 
we never had sufficient reason for making. Just as we 
were misled into untenable ideas of the aether through 
trusting to an analogy with the material ocean, so we 
have been misled into untenable ideas of the attributes 
of the microscopic elements of world-structure through 
trusting to analogy with gross particles. 

A New Epistemology. The principle of indeterminacy 
is epistemological. It reminds us once again that the 
world of physics is a world contemplated from within 
surveyed by appliances which are part of it and subject 
to its laws. What the world might be deemed like if 
probed in some supernatural manner by appliances not 
furnished by itself we do not profess to know. 



226 THE NEW QUANTUM THEORY 

There is a doctrine well known to philosophers that 
the moon ceases to exist when no one is looking at it. 
I will not discuss the doctrine since I have not the least 
idea what is the meaning of the word existence when 
used in this connection. At any rate the science of as- 
tronomy has not been based on this spasmodic kind of 
moon. In the scientific world (which has to fulfil func- 
tions less vague than merely existing) there is a moon 
which appeared on the scene before the astronomer; it 
reflects sunlight when no one sees it; it has mass when 
no one is measuring the mass; it is distant 240,000 miles 
from the earth when no one is surveying the distance; 
and it will eclipse the sun in 1999 even if the human race 
has succeeding in killing itself off before that date. The 
moon — the scientific moon — has to play the part of a 
continuous causal element in a world conceived to be all 
causally interlocked. 

What should we regard as a complete description of 
this scientific world? We must not introduce anything 
like velocity through aether, which is meaningless since 
it is not assigned any causal connection with our ex- 
perience. On the other hand we cannot limit the de- 
scription to the immediate data of our own spasmodic 
observations. The description should include nothing 
that is unobservable but a great deal that is actually 
unobserved. Virtually we postulate an infinite army of 
watchers and measurers. From moment to moment they 
survey everything that can be surveyed and measure 
everything that can be measured by methods which we 
ourselves might conceivably employ. Everything they 
measure goes down as part of the complete description 
of the scientific world. We can, of course, introduce 
derivative descriptions, words expressing mathematical 
combinations of the immediate measures which may give 



A NEW EPISTEMOLOGY 227 

greater point to the description — so that we may not 
miss seeing the wood for the trees. 

By employing the known physical laws expressing 
the uniformities of Nature we can to a large extent 
dispense with this army of watchers. We can afford to 
let the moon out of sight for an hour or two and deduce 
where it has been in the meantime. But when I assert 
that the moon (which I last saw in the west an hour ago) 
is now setting, I assert this not as my deduction but as 
a true fact of the scientific world. I am still postulating 
the imaginary watcher; I do not consult him, but I 
retain him to corroborate my statement if it is chal- 
lenged. Similarly, when we say that the distance of 
Sirius is 50 billion miles we are not giving a merely con- 
ventional interpretation to its measured parallax; we in- 
tend to give it the same status in knowledge as if some- 
one had actually gone through the operation of laying 
measuring rods end to end and counted how many were 
needed to reach to Sirius; and we should listen patiently 
to anyone who produced reasons for thinking that our 
deductions did not correspond to the "real facts", i.e. 
the facts as known to our army of measurers. If we 
happen to make a deduction which could not conceivably 
be corroborated or disproved by these diligent measur- 
ers, there is no criterion of its truth or falsehood and it 
is thereby a meaningless deduction. 

This theory of knowledge is primarily intended to 
apply to our macroscopic or large-scale survey of the 
physical world, but it has usually been taken for granted 
that it is equally applicable to a microscopic study. We 
have at last realised the disconcerting fact that though 
it applies to the moon it does not apply to the 
electron. 

It does not hurt the moon to look at it. There is no 



228 THE NEW QUANTUM THEORY 

inconsistency in supposing it to have been under the 
surveillance of relays of watchers whilst we were asleep. 
But it is otherwise with an electron. At certain times, 
viz. when it is interacting with a quantum, it might be 
detected by one of our watchers; but between whiles it 
virtually disappears from the physical world, having no 
interaction with it. We might arm our observers with 
flash-lamps to keep a more continuous watch on its 
doings; but the trouble is that under the flashlight it 
will not go on doing what it was doing in the dark. 
There is a fundamental inconsistency in conceiving the 
microscopic structure of the physical world to be under 
continuous survey because the surveillance would itself 
wreck the whole machine. 

I expect that at first this will sound to you like a 
merely dialectical difficulty. But there is much more in 
it than that. The deliberate frustration of our efforts to 
bring knowledge of the microscopic world into orderly 
plan, is a strong hint to alter the plan. 

It means that we have been aiming at a false ideal of 
a complete description of the world. There has not yet 
been time to make serious search for a new epistemology 
adapted to these conditions. It has become doubtful 
whether it will ever be possible to construct a physical 
world solely out of the knowable — the guiding principle 
in our macroscopic theories. If it is possible, it involves 
a great upheaval of the present foundations. It seems 
more likely that we must be content to admit a mixture 
of the knowable and unknowable. This means a denial 
of determinism, because the data required for a pre- 
diction of the future will include the unknowable ele- 
ments of the past. I think it was Heisenberg who said, 
u The question whether from a complete knowledge of 
the past we can predict the future, does not arise because 



A NEW EPISTEMOLOGY 229 

a complete knowledge of the past involves a self-con- 
tradiction." 

It is only through a quantum action that the outside 
world can interact with ourselves and knowledge of it 
can reach our minds. A quantum action may be the 
means of revealing to us some fact about Nature, but 
simultaneously a fresh unknown is implanted in the womb 
of Time. An addition to knowledge is won at the ex- 
pense of an addition to ignorance. It is hard to empty 
the well of Truth with a leaky bucket. 



Chapter XI 

WORLD BUILDING 

We have an intricate task before us. We are going to 
build a World — a physical world which will give a 
shadow performance of the drama enacted in the world 
of experience. We are not very expert builders as yet; 
and you must not expect the performance to go off 
without a hitch or to have the richness of detail which a 
critical audience might require. But the method about 
to be described seems to give the bold outlines; doubt- 
less we have yet to learn other secrets of the craft of 
world building before we can complete the design. 

The first problem is the building material. I remem- 
ber that as an impecunious schoolboy I used to read 
attractive articles on how to construct wonderful con- 
trivances out of mere odds and ends. Unfortunately 
these generally included the works of an old clock, a 
few superfluous telephones, the quicksilver from a 
broken barometer, and other oddments which happened 
not to be forthcoming in my lumber room. I will try 
not to let you down like that. I cannot make the world 
out of nothing, but I will demand as little specialised 
material as possible. Success in the game of World 
Building consists in the greatness of the contrast 
between the specialised properties of the completed 
structure and the unspecialised nature of the basal 
material. 

Relation Structure. We take as building material rela- 
tions and relata. The relations unite the relata; the 
relata are the meeting points of the relations. The one 

230 



RELATION STRUCTURE 231 

is unthinkable apart from the other. I do not think that 
a more general starting-point of structure could be 
conceived. 

To distinguish the relata from one another we assign 
to them monomarks. The monomark consists of four 
numbers ultimately to be called "co-ordinates". But 
co-ordinates suggest space and geometry and as yet there 
is no such thing in our scheme; hence for the present 
we shall regard the four identification numbers as no 
more than an arbitrary monomark. Why four numbers? 
We use four because it turns out that ultimately the 
structure can be brought into better order that way; 
but we do not know why this should be so. We have 
got so far as to understand that if the relations insisted 
on a threefold or a fivefold ordering it would be much 
more difficult to build anything interesting out of them; 
but that is perhaps an insufficient excuse for the 
special assumption of fourfold order in the primitive 
material. 

The relation between two human individuals in its 
broadest sense comprises every kind of connection or 
comparison between them — consanguinity, business trans- 
actions, comparative stature, skill at golf — any kind of 
description in which both are involved. For generality 
we shall suppose that the relations in our world-material 
are likewise composite and in no way expressible in nu- 
merical measure. Nevertheless there must be some kind 
of comparability or likeness of relations, as there is in 
the relations of human individuals; otherwise there 
would be nothing more to be said about the world than 
that everything in it was utterly unlike everything else. 
To put it another way, we must postulate not only rela- 
tions between the relata but some kind of relation of 
likeness between some of the relations. The slightest 



232 WORLD BUILDING 

concession in this direction will enable us to link the 
whole into a structure. 

We assume then that, considering a relation between 
two relata, it will in general be possible to pick out two 
other relata close at hand which stand to one another 
in a "like" relation. By "like" I do not mean "like in 
every respect", but like in respect to one of the aspects 
of the composite relation. How is the particular aspect 
selected? If our relata were human individuals different 
judgments of likeness would be made by the geneal- 
ogist, the economist, the psychologist, the sportsman, 
etc.; and the building of structure would here diverge 
along a number of different lines. Each could build his 
own world-structure from the common basal material 
of humanity. There is no reason to deny that a similar 
diversity of worlds could be built out of our postulated 
material. But all except one of these worlds will be 
stillborn. Our labour will be thrown away unless the 
world we have built is the one which the mind chooses 
to vivify into a world of experience. The only definition 
we can give of the aspect of the relations chosen for the 
criterion of likeness, is that it is the aspect which will 
ultimately be concerned in the getting into touch of mind 
w T ith the physical world. But that is beyond the province 
of physics. 

This one-to-one correspondence of "likeness" is only 
supposed to be definite in the limit when the relations 
are very close together in the structure. Thus we avoid 
any kind of comparison at a distance which is as 
objectionable as action at a distance. Let me confess at 
once that I do not know what I mean here by "very 
close together". As yet space and time have not been 
built. Perhaps we might say that only a few of the 
relata possess relations whose comparability to the first 



RELATION STRUCTURE 



233 



is definite, and take the definiteness of the comparability 
as the criterion of contiguity. I hardly know. The 
building at this point shows some cracks, but I think it 
should not be beyond the resources of the mathematical 
logician to cement them up. We should also arrange at 
this stage that the monomarks are so assigned as to give 
an indication of contiguity. 




Fig. 7 

Let us start with a relatum A and a relation AP 
radiating from it. Now step to a contiguous relatum 
B and pick out the "like" relation BQ. Go on to 
another contiguous relatum C and pick out the relation 
CR which is like BQ. (Note that since C is farther 
from A than from B } the relation at C which is like 
AP is not so definite as the relation which is like BQ.) 
Step by step we may make the comparison round a 
route AEFA which returns to the starting-point. There 
is nothing to ensure that the final relation AP' which 



234 WORLD BUILDING 

has, so to speak, been carried round the circuit will be 
the relation AP with which we originally started. 

We have now two relations AP, AP' radiating from 
the first relatum, their difference being connected with 
a certain circuit in the world AEFA. The loose ends of 
the relations P and P have their monomarks, and we 
can take the difference of the monomarks (i.e. the 
difference of the identification numbers comprised in 
them) as the code expression for the change introduced 
by carrying AP round the circuit. As we vary the circuit 
and the original relation, so the change PP' varies; and 
the next step is to find a mathematical formula express- 
ing this dependence. There are virtually four things to 
connect, the circuit counting double since, for example, 
a rectangular circuit would be described by specifying 
two sides. Each of them has to be specified by four 
identification numbers (either monomarks or derived 
from monomarks) ; consequently, to allow for all com- 
binations, the required mathematical formula contains 
4 4 or 256 numerical coefficients. These coefficients give 
a numerical measure of the structure surrounding the 
initial relatum. 

This completes the first part of our task to introduce 
numerical measure of structure into the basal material. 
The method is not so artificial as it appears at first sight. 
Unless we shirk the problem by putting the desired 
physical properties of the world directly into the original 
relations and relata, we must derive them from the 
structural interlocking of the relations; and such 
interlocking is naturally traced by following circuits 
among the relations. The axiom of comparability of 
contiguous relations only discriminates between like 
and unlike, and does not initially afford any means 
of classifying various decrees and kinds of unlikeness; 



RELATION STRUCTURE 235 

but we have found a means of specifying the kind 
of unlikeness of AP and AP' by reference to a circuit 
which "transforms" one into the other. Thus we have 
built a quantitative study of diversity on a definition of 
similarity. 

The numerical measures of structure will be dependent 
on, and vary according to, the arbitrary code of mono- 
marks used for the identification of relata. This, how- 
ever, renders them especially suitable for building the 
ordinary quantities of physics. When the monomarks 
become co-ordinates of space and time the arbitrary 
choice of the code will be equivalent to the arbitrary 
choice of a frame of space and time; and it is in accord- 
ance with the theory of relativity that the measures of 
structure and the physical quantities to be built from 
them should vary with the frame of space and time. 
Physical quantities in general have no absolute value, 
but values relative to chosen frames of reference or 
codes of monomarks. 

We have now fashioned our bricks from the primitive 
clay and the next job is to build with them. The 256 
measures of structure varying from point to point of 
the world are somewhat reduced in number when dupli- 
cates are omitted; but even so they include a great deal 
of useless lumber which we do not require for the 
building. That seems to have worried a number of the 
most eminent physicists; but I do not quite see why. 
Ultimately it is the mind that decides what is lumber — 
which part of our building will shadow the things of 
common experience, and which has no such counterpart. 
It is no part of our function as purveyors of building 
material to anticipate what will be chosen for the 
palace of the mind. The lumber will now be dropped as 
irrelevant in the further operations, but I do not agree 



236 WORLD BUILDING 

with those who think it a blemish on the theory that 
the lumber should ever have appeared in it. 

By adding together certain of the measures of struc- 
ture in a symmetrical manner and by ignoring others 
we reduce the really important measures to 16.* These 
can be divided into 10 forming a symmetrical scheme 
and 6 forming an antisymmetrical scheme. This is the 
great point of bifurcation of the world. 

Symmetrical coefficients (10). Out of these we find it 
possible to construct Geometry and Mechanics. They 
are the ten potentials of Einstein (g,J). We derive 
from them space, time, and the world-curvatures re- 
presenting the mechanical properties of matter, viz. 
momentum, energy, stress, etc. 

Antisymmetrical coefficients (6). Out of these we con- 
struct Electromagnetism. They are the three com- 
ponents of electric intensity and three components of 
magnetic force. We derive electric and magnetic 
potential, electric charge and current, light and other 
electric waves. 

We do not derive the laws and phenomena of 
atomicity. Our building operation has somehow been 
too coarse to furnish the microscopic structure of the 
world, so that atoms, electrons and quanta are at present 
beyond our skill. 

But in regard to what is called field-physics the 
construction is reasonably complete. The metrical, 
gravitational and electromagnetic fields are all included. 
We build the quantities enumerated above; and they 
obey the great laws of field-physics in virtue of the way 
in which they have been built. That is the special fea- 
ture; the field laws — conservation of energy, mass, mo- 

* Mathematically we contract the original tensor of the fourth rank 
to one of the second rank. 



IDENTICAL LAWS 237 

mentum and of electric charge, the law of gravitation, 
Maxwell's equations — are not controlling laws.* They 
are truisms. Not truisms when approached in the way 
the mind looks out on the world, but truisms when we 
encounter them in a building up of the world from a 
basal structure. I must try to make clear our new 
attitude to these laws. 

Identical Laws. Energy momentum and stress, which 
we have identified with the ten principal curvatures of 
the world, are the subject of the famous laws of con- 
servation of energy and momentum. Granting that the 
identification is correct, these laws are mathematical 
identities. Violation of them is unthinkable. Perhaps 
I can best indicate their nature by an analogy. 

An aged college Bursar once dwelt secluded in his 
rooms devoting himself entirely to accounts. He realised 
the intellectual and other activities of the college only 
as they presented themselves in the bills. He vaguely 
conjectured an objective reality at the back of it all — 
some sort of parallel to the real college — though he 
could only picture it in terms of the pounds, shillings 
and pence which made up what he would call "the 
commonsense college of everyday experience". The 
method of account-keeping had become inveterate habit 
handed down from generations of hermit-like bursars; 
he accepted the form of accounts as being part of the 
nature of things. But he was of a scientific turn and he 
wanted to learn more about the college. One day in 
looking over his books he discovered a remarkable law. 

* One law commonly grouped with these, viz. the law of pondero- 
motive force of the electric field, is not included. It seems to be impos- 
sible to get at the origin of this law without tackling electron structure 
which is beyond the scope of our present exercise in world-building. 



238 WORLD BUILDING 

For every item on the credit side an equal item appeared 
somewhere else on the debit side. "Ha I" said the 
Bursar, "I have discovered one of the great laws con- 
trolling the college. It is a perfect and exact law of the 
real world. Credit must be called plus and debit minus; 
and so we have the law of conservation of £ s. d. This 
is the true way to find out things, and there is no limit 
to what may ultimately be discovered by this scientific 
method. I will pay no more heed to the superstitions 
held by some of the Fellows as to a beneficent spirit 
called the King or evil spirits called the University 
Commissioners. I have only to go on in this way and 
I shall succeed in understanding why prices are always 
going up." 

I have no quarrel with the Bursar for believing that 
scientific investigation of the accounts is a road to exact 
(though necessarily partial) knowledge of the reality 
behind them. Things may be discovered by this method 
which go deeper than the mere truism revealed by his 
first effort. In any case his life is especially concerned 
with accounts and it is proper that he should discover 
the laws of accounts whatever their nature. But I would 
point out to him that a discovery of the overlapping of 
the different aspects in which the realities of the college 
present themselves in the world of accounts, is not a 
discovery of the laws controlling the college; that he 
has not even begun to find the controlling laws. The 
college may totter but the Bursar's accounts still balance. 

The law of conservation of momentum and energy 
results from the overlapping of the different aspects in 
which the "non-emptiness of space" presents itself to 
our practical experience. Once again we find that a 
fundamental law of physics is no controlling law but a 
"put-up job" as soon as we have ascertained the nature 



SELECTIVE INFLUENCE OF THE MIND 239 

of that which is obeying it. We can measure certain 
forms of energy with a thermometer, momentum with 
a ballistic pendulum, stress with a manometer. Com- 
monly we picture these as separate physical entities 
whose behaviour towards each other is controlled by 
a law. But now the theory is that the three instruments 
measure different but slightly overlapping aspects of a 
single physical condition, and a law connecting their 
measurements is of the same tautological type as a "law" 
connecting measurements with a metre-rule and a foot- 
rule. 

I have said that violation of these laws of conserva- 
tion is unthinkable. Have we then found physical laws 
which will endure for all time unshaken by any future 
revolution? But the proviso must be remembered, 
"granting that the identification [of their subject 
matter] is correct". The law itself will endure as long 
as two and two make four; but its practical importance 
depends on our knowing that which obeys it. We 
think we have this knowledge, but do not claim in- 
fallibility in this respect. From a practical point of view 
the law would be upset, if it turned out that the thing 
conserved was not that which we are accustomed to 
measure with the above-mentioned instruments but 
something slightly different. 

Selective Influence of the Mind. This brings us very 
near to the problem of bridging the gulf between the 
scientific world and the world of everyday experience. 
The simpler elements of the scientific world have no 
immediate counterparts in everyday experience; we 
use them to build things which have counterparts. 
Energy, momentum and stress in the scientific world 
shadow well-known features of the familiar world. 



240 WORLD BUILDING 

I feel stress in my muscles; one form of energy gives me 
the sensation of warmth; the ratio of momentum to mass 
is velocity, which generally enters into my experience 
as change of position of objects. When I say that I feel 
these things I must not forget that the feeling, in so far 
as it is located in the physical world at all, is not in the 
things themselves but in a certain corner of my brain. 
In fact, the mind has also invented a craft of world- 
building; its familiar world is built not from the dis- 
tribution of relata and relations but by its own peculiar 
interpretation of the code messages transmitted along 
the nerves into its sanctum. 

Accordingly we must not lose sight of the fact that 
the world which physics attempts to describe arises 
from the convergence of two schemes of world-building. 
If we look at it only from the physical side there is 
inevitably an arbitrariness about the building. Given 
the bricks — the 16 measures of world-structure — there 
are all sorts of things we might build. Or we might 
take up again some of the rejected lumber and build a 
still wider variety of things. But we do not build 
arbitrarily; we build to order. The things we build have 
certain remarkable properties; they have these pro- 
perties in virtue of the way they are built, but they also 
have them because such properties were ordered. There 
is a general description which covers at any rate most 
of the building operations needed in the construction 
of the physical world; in mathematical language the 
operation consists in Hamiltonian differentiation of an 
invariant function of the 16 measures of structure. I do 
not think that there is anything in the basal relation- 
structure that cries out for this special kind of com- 
bination; the significance of this process is not in 
inorganic nature. Its significance is that it corresponds 



SELECTIVE INFLUENCE OF THE MIND 241 

to an outlook adopted by the mind for its own reasons; 
and any other building process would not converge to 
the mental scheme of world-building. The Hamiltonian 
derivative has just that kind of quality which makes it 
stand out in our minds as an active agent against a 
passive extension of space and time; and Hamiltonian 
differentiation is virtually the symbol for creation of an 
active world out of the formless background. Not once 
in the dim past, but continuously by conscious mind is 
the miracle of the Creation wrought. 

By following this particular plan of building we 
construct things which satisfy the law of conservation, 
that is to say things which are permanent. The law of 
conservation is a truism for the things which satisfy it; 
but its prominence in the scheme of law of the physical 
world is due to the mind having demanded permanence. 
We might have built things which do not satisfy this 
law. In fact we do build one very important thing 
"action" which is not permanent; in respect to "action" 
physics has taken the bit in her teeth, and has insisted 
on recognising this as the most fundamental thing of all, 
although the mind has not thought it worthy of a place 
in the familiar world and has not vivified it by any 
mental image or conception. You will understand that 
the building to which I refer is not a shifting about of 
material; it is like building constellations out of stars. 
The things which we might have built but did not, are 
there just as much as those we did build. What we have 
called building is rather a selection from the patterns 
that weave themselves. 

The element of permanence in the physical world, 
which is familiarly represented by the conception of 
substance, is essentially a contribution of the mind to 
the plan of building or selection. We can see this 



242 WORLD BUILDING 

selective tendency at work in a comparatively simple 
problem, viz. the hydrodynamical theory of the ocean. 
At first sight the problem of what happens when the 
water is given some initial disturbance depends solely on 
inorganic laws; nothing could be more remote from the 
intervention of conscious mind. In a sense this is true; 
the laws of matter enable us to work out the motion 
and progress of the different portions of the water; and 
there, so far as the inorganic world is concerned, the 
problem might be deemed to end. But actually in 
hydrodynamical textbooks the investigation is diverted 
in a different direction, viz. to the study of the motions 
of waves and wave-groups. The progress of a wave is 
not progress of any material mass of water, but of a 
form which travels over the surface as the water heaves 
up and down; again the progress of a wave-group is not 
the progress of a wave. These forms have a certain 
degree of permanence amid the shifting particles of 
water. Anything permanent tends to become dignified 
with an attribute of substantiality. An ocean traveller 
has even more vividly the impression that the ocean is 
made of waves than that it is made of water.* Ulti- 
mately it is this innate hunger for permanence in our 
minds which directs the course of development of 
hydrodynamics, and likewise directs the world-building 
out of the sixteen measures of structure. 

Perhaps it will be objected that other things besides 
mind can appreciate a permanent entity such as mass; 
a weighing machine can appreciate it and move a 
pointer to indicate how much mass there is. I do not 
think that is a valid objection. In building the physical 
world we must of course build the measuring appliances 

* This was not intended to allude to certain consequential effects of 
the waves; it is true, I think, of the happier impressions of the voyage. 



SELECTIVE INFLUENCE OF THE MIND 243 

which are part of it; and the measuring appliances 
result from the plan of building in the same way as the 
entities which they measure. If, for example, we had 
used some of the "lumber" to build an entity x, we 
could presumably construct from the same lumber an 
appliance for measuring x. The difference is this — if the 
pointer of the weighing machine is reading 5 lbs. a 
human consciousness is in a mysterious way (not yet 
completely traced) aware of the fact, whereas if the 
measuring appliance for x reads 5 units no human mind 
is aware of it. Neither x nor the appliance for measur- 
ing x have any interaction with consciousness. Thus the 
responsibility for the fact that the scheme of the scientific 
world includes mass but excludes x rests ultimately with 
the phenomena of consciousness. 

Perhaps a better way of expressing this selective 
influence of mind on the laws of Nature is to say that 
values are created by the mind. All the "light and shade" 
in our conception of the world of physics comes in this 
way from the mind, and cannot be explained without 
reference to the characteristics of consciousness. 

The world which we have built from the relation- 
structure is no doubt doomed to be pulled about a good 
deal as our knowledge progresses. The quantum theory 
shows that some radical change is impending. But I 
think that our building exercise has at any rate widened 
our minds to the possibilities and has given us a different 
orientation towards the idea of physical law. The points 
which I stress are: 

Firstly, a strictly quantitative science can arise from 
a basis which is purely qualitative. The comparability 
that has to be assumed axiomatically is a merely quali- 
tative discrimination of likeness and unlikeness. 

Secondly, the laws which we have hitherto regarded 



244 WORLD BUILDING 

as the most typical natural laws are of the nature of 
truisms, and the ultimate controlling laws of the basal 
structure (if there are any) are likely to be of a differ- 
ent type from any yet conceived. 

Thirdly, the mind has by its selective power fitted 
the processes of Nature into a frame of law of a pattern 
largely of its own choosing; and in the discovery of this 
system of law the mind may be regarded as regaining 
from Nature that which the mind has put into 
Nature. 

Three Types of Law. So far as we are able to judge, the 
laws of Nature divide themselves into three classes: 
(i) identical laws, (2) statistical laws, (3) transcenden- 
tal laws. We have just been considering the identical 
laws, i.e. the laws obeyed as mathematical identities in 
virtue of the way in which the quantities obeying them 
are built. They cannot be regarded as genuine laws of 
control of the basal material of the world. Statistical 
laws relate to the behaviour of crowds, and depend on 
the fact that although the behaviour of each individual 
may be extremely uncertain average results can be 
predicted with confidence. Much of the apparent uni- 
formity of Nature is a uniformity of averages. Our 
gross senses only take cognisance of the average effect of 
vast numbers of individual particles and processes; and 
the regularity of the average might well be compatible 
with a great degree of lawlessness of the individual. I do 
not think it is possible to dismiss statistical laws (such 
as the second law of thermodynamics) as merely mathe- 
matical adaptations of the other classes of law to certain 
practical problems. They involve a peculiar element 
of their own connected with the notion of a priori proba- 
bility; but we do not yet seem able to find a place for 



THREE TYPES OF LAW 245 

this in any of the current conceptions of the world sub- 
stratum. 

If there are any genuine laws of control of the physical 
world they must be sought in the third group — the 
transcendental laws. The transcendental laws comprise 
all those which have not become obvious identities im- 
plied in the scheme of world-building. They are con- 
cerned with the particular behaviour of atoms, electrons 
and quanta — that is to say, the laws of atomicity of 
matter, electricity and action. We seem to be mak- 
ing some progress towards formulating them, but it is 
clear that the mind is having a much harder struggle to 
gain a rational conception of them than it had with the 
classical field-laws. We have seen that the field-laws, 
especially the laws of conservation, are indirecdy imposed 
by the mind which has, so to speak, commanded a plan of 
world-building to satisfy them. It is a natural suggestion 
that the greater difficulty in elucidating the transcenden- 
tal laws is due to the fact that we are no longer engaged 
in recovering from Nature what we have ourselves put 
into Nature, but are at last confronted with its own in- 
trinsic system of government. But I scarcely know what 
to think. We must not assume that the possible develop- 
ments of the new attitude towards natural law have been 
exhausted in a few short years. It may be that the laws 
of atomicity, like the laws of conservation, arise only in 
the presentation of the world to us and can be recognised 
as identities by some extension of the argument we have 
followed. But it is perhaps as likely that after we have 
cleared away all the superadded laws which arise solely 
in our mode of apprehension of the world about us, there 
will be left an external world developing under genuine 
laws of control. 

At present we can notice the contrast that the laws 



246 WORLD BUILDING 

which we now recognise as man-made are characterised 
by continuity, whereas the laws to which the mind as 
yet lays no claim are characterised by atomicity. The 
quantum theory with its avoidance of fractions and 
insistence on integral units seems foreign to any scheme 
which we should be likely subconsciously to have im- 
posed as a frame for natural phenomena. Perhaps our 
final conclusion as to the world of physics will resemble 
Kronecker's view of pure mathematics. 

u God made the integers, all else is the work of man."* 

* Die ganzen Zahlen hat Gott gemacht; alles anderes ist Menschen- 
werk. 



Chapter XII 
POINTER READINGS 

Familiar Conceptions and Scientific Symbols. We have 
said in the Introduction that the raw material of the 
scientific world is not borrowed from the familiar world. 
It is only recently that the physicist has deliberately cut 
himself adrift from familiar conceptions. He did not 
set out to discover a new world but to tinker with the 
old. Like everyone else he started with the idea that 
things are more or less what they seem, and that our 
vivid impression of our environment may be taken as 
a basis to work from. Gradually it has been found that 
some of its most obvious features must be rejected. We 
learn that instead of standing on a firm immovable earth 
proudly rearing our heads towards the vault of heaven, 
we are hanging by our feet from a globe careering 
through space at a great many miles a second. But this 
new knowledge can still be grasped by a rearrangement 
of familiar conceptions. I can picture to myself quite 
vividly the state of affairs just described; if there is any 
strain, it is on my credulity, not on my powers of con- 
ception. Other advances of knowledge can be accommo- 
dated by that very useful aid to comprehension — "like 
this only more so". For example, if you think of some- 
thing like a speck of dust only more so you have the 
atom as it was conceived up to a fairly recent date. 

In addition to the familiar entities the physicist had 
to reckon with mysterious agencies such as gravitation 
or electric force; but this did not disturb his general 
outlook. We cannot say what electricity is "like"j but 

247 



248 POINTER READINGS 

at first its aloofness was not accepted as final. It was 
taken to be one of the main aims of research to discover 
how to reduce these agencies to something describable 
in terms of familiar conceptions — in short to "explain" 
them. For example, the true nature of electric force 
might be some kind of displacement of the aether. 
(Aether was at that time a familiar conception — like 
some extreme kind of matter only more so.) Thus 
there grew up a waiting-list of entities which should 
one day take on their rightful relation to conceptions 
of the familiar world. Meanwhile physics had to 
treat them as best it could without knowledge of their 
nature. 

It managed surprisingly well. Ignorance of the nature 
of these entities was no bar to successful prediction of 
behaviour. We gradually awoke to the fact that the 
scheme of treatment of quantities on the waiting-list 
was becoming more precise and more satisfying than 
our knowledge of familiar things. Familiar conceptions 
did not absorb the waiting-list, but the waiting-list 
began to absorb familiar conceptions. Aether, after 
being in turn an elastic solid, a jelly, a froth, a con- 
glomeration of gyrostats, was denied a material and 
substantial nature and put back on the waiting-list. It 
was found that science could accomplish so much with 
entities whose nature was left in suspense that it began 
to be questioned whether there was any advantage in 
removing the suspense. The crisis came when we began 
to construct familiar entities such as matter and light 
out of things on the waiting-list. Then at last it was seen 
that the linkage to familiar concepts should be through 
the advanced constructs of physics and not at the be- 
ginning of the alphabet. We have suffered, and we still 
suffer, from expectations that electrons and quanta must 



SCIENTIFIC SYMBOLS 249 

be in some fundamental respects like materials or forces 
familiar in the workshop — that all we have got to do is 
to imagine the usual kind of thing on an infinitely smaller 
scale. It must be our aim to avoid such prejudgments, 
which are surely illogical; and since we must cease to 
employ familiar concepts, symbols have become the only 
possible alternative. 

The synthetic method by which we build up from 
its own symbolic elements a world which will imitate 
the actual behaviour of the world of familiar experience 
is adopted almost universally in scientific theories. Any 
ordinary theoretical paper in the scientific journals 
tacitly assumes that this approach is adopted. It has 
proved to be the most successful procedure; and it is the 
actual procedure underlying the advances set forth in 
the scientific part of this book. But I would not claim 
that no other way of working is admissible. We agree 
that at the end of the synthesis there must be a linkage 
to the familiar world of consciousness, and we are not 
necessarily opposed to attempts to reach the physical 
world from that end. From the point of view of philo- 
sophy it is desirable that this entrance should be 
explored, and it is conceivable that it may be fruitful 
scientifically. If I have rightly understood Dr. White- 
head's philosophy, that is the course which he takes. It 
involves a certain amount of working backwards (as 
we should ordinarily describe it) ; but his method of 
"extensive abstraction" is intended to overcome some 
of the difficulties of such a procedure. I am not qualified 
to form a critical judgment of this work, but in principle 
it appears highly interesting. Although this book may 
in most respects seem diametrically opposed to Dr. 
Whitehead's widely read philosophy of Nature, I think 
it would be truer to regard him as an ally who from the 



250 POINTER READINGS 

opposite side of the mountain is tunnelling to meet his 
less philosophically minded colleagues. The important 
thing is not to confuse the two entrances. 

Nature of Exact Science. One of the characteristics of 
physics is that it is an exact science, and I have generally 
identified the domain of physics with the domain of 
exact science. Strictly speaking the two are not synony- 
mous. We can imagine a science arising which has no 
contact with the usual phenomena and laws of physics, 
which yet admits of the same kind of exact treatment. 
It is conceivable that the Mendelian theory of heredity 
may grow into an independent science of this kind, for 
it would seem to occupy in biology the same position 
that the atomic theory occupied in chemistry a hundred 
years ago. The trend of the theory is to analyse com- 
plex individuals into "unit characters". These are like 
indivisible atoms with affinities and repulsions; their 
matings are governed by the same laws of chance which 
play so large a part in chemical thermodynamics; and 
numerical statistics of the characters of a population are 
predictable in the same way as the results of a chemical 
reaction. 

Now the effect of such a theory on our philosophical 
views of the significance of life does not depend on 
whether the Mendelian atom admits of a strictly physical 
explanation or not. The unit character may be contained 
in some configuration of the physical molecules of the 
carrier, and perhaps even literally correspond to a chem- 
ical compound; or it may be something superadded which 
is peculiar to living matter and is not yet comprised in 
the schedule of physical entities. That is a side-issue. 
We are drawing near to the great question whether there 
is any domain of activity — of life, of consciousness, of 



NATURE OF EXACT SCIENCE 251 

deity — which will not be engulfed by the advance of 
exact science; and our apprehension is not directed 
against the particular entities of physics but against all 
entities of the category to which exact science can apply. 
For exact science invokes, or has seemed to invoke, a 
type of law inevitable and soulless against which the 
human spirit rebels. If science finally declares that man 
is no more than a fortuitous concourse of atoms, the 
blow will not be softened by the explanation that the 
atoms in question are the Mendelian unit characters 
and not the material atoms of the chemist. 

Let us then examine the kind of knowledge which is 
handled by exact science. If we search the examination 
papers in physics and natural philosophy for the more 
intelligible questions we may come across one beginning 
something like this: "An elephant slides down a 
grassy hillside. . . ." The experienced candidate knows 
that he need not pay much attention to this; it is only 
put in to give an impression of realism. He reads on: 
"The mass of the elephant is two tons." Now we are 
getting down to business; the elephant fades out of the 
problem and a mass of two tons takes its place. What 
exactly is this two tons, the real subject-matter of the 
problem? It refers to some property or condition which 
we vaguely describe as "ponderosity" occurring in a 
particular region of the external world. But we shall not 
get much further that way; the nature of the external 
world is inscrutable, and we shall only plunge into a 
quagmire of indescribables. Never mind what two tons 
refers to; what is it? How has it actually entered in so 
definite a way into our experience? Two tons is the 
reading of the pointer when the elephant was placed 
on a weighing-machine. Let us pass on. "The slope 
of the hill is 6o°." Now the hillside fades out of the 



252 POINTER READINGS 

problem and an angle of 6o° takes its place. What is 
6o°? There is no need to struggle with mystical con- 
ceptions of direction; 6o° is the reading of a plumb-line 
against the divisions of a protractor. Similarly for the 
other data of the problem. The softly yielding turf on 
which the elephant slid is replaced by a coefficient of 
friction, which though perhaps not direcdy a pointer 
reading is of kindred nature. No doubt there are more 
roundabout ways used in practice for determining the 
weights of elephants and the slopes of hills, but these 
are justified because it is known that they give the same 
results as direct pointer readings. 

And so we see that the poetry fades out of the prob- 
lem, and by the time the serious application of exact 
science begins we are left with only pointer readings. 
If then only pointer readings or their equivalents are 
put into the machine of scientific calculation, how can 
we grind out anything but pointer readings? But that 
is just what we do grind out. The question presumably 
was to find the time of descent of the elephant, and the 
answer is a pointer reading on the seconds' dial of our 
watch. 

The triumph of exact science in the foregoing problem 
consisted in establishing a numerical connection between 
the pointer reading of the weighing-machine in one 
experiment on the elephant and the pointer reading of 
the watch in another experiment. And when we examine 
critically other problems of physics we find that this is 
typical. The whole subject-matter of exact science 
consists of pointer readings and similar indications. 
We cannot enter here into the definition of what are 
to be classed as similar indications. The observation of 
approximate coincidence of the pointer with a scale- 
division can generally be extended to include the 



NATURE OF EXACT SCIENCE 253 

observation of any kind of coincidence — or, as it is 
usually expressed in the language of the general rela- 
tivity theory, an intersection of world-lines. The 
essential point is that, although we seem to have very 
definite conceptions of objects in the external world, 
those conceptions do not enter into exact science and 
are not in any way confirmed by it. Before exact science 
can begin to handle the problem they must be replaced 
by quantities representing the results of physical meas- 
urement. 

Perhaps you will object that although only the pointer 
readings enter into the actual calculation it would make 
nonsense of the problem to leave out all reference to 
anything else. The problem necessarily involves some 
kind of connecting background. It was not the pointer 
reading of the weighing-machine that slid down the 
hill! And yet from the point of view of exact science the 
thing that really did descend the hill can only be de- 
scribed as a bundle of pointer readings. (It should be 
remembered that the hill also has been replaced by 
pointer readings, and the sliding down is no longer an 
active adventure but a functional relation of space and 
time measures.) The word elephant calls up a certain 
association of mental impressions, but it is clear that 
mental impressions as such cannot be the subject 
handled in the physical problem. We have, for example, 
an impression of bulkiness. To this there is presumably 
some direct counterpart in the external world, but that 
counterpart must be of a nature beyond our appre- 
hension, and science can make nothing of it. Bulkiness 
enters into exact science by yet another substitution; 
we replace it by a series of readings of a pair of cali- 
pers. Similarly the greyish black appearance in our 
mental impression is replaced in exact science by the read- 



254 POINTER READINGS 

ings of a photometer for various wave-lengths of light. 
And so on until all the characteristics of the elephant 
are exhausted and it has become reduced to a schedule 
of measures. There is always the triple correspond- 
ence — 

(a) a mental image, which is in our minds and not in 
the external world; 

(b) some kind of counterpart in the external world, 
which is of inscrutable nature; 

(<:) a set of pointer readings, which exact science can 
study and connect with other pointer readings. 

And so we have our schedule of pointer readings 
ready to make the descent. And if you still think that 
this substitution has taken away all reality from the 
problem, I am not sorry that you should have a foretaste 
of the difficulty in store for those who hold that exact 
science is all-sufficient for the description of the universe 
and that there is nothing in our experience which cannot 
be brought within its scope. 

I should like to make it clear that the limitation of 
the scope of physics to pointer readings and the like is 
not a philosophical craze of my own but is essentially 
the current scientific doctrine. It is the outcome of a 
tendency discernible far back in the last century but 
only formulated comprehensively with the advent of 
the relativity theory. The vocabulary of the physicist 
comprises a number of words such as length, angle, 
velocity, force, potential, current, etc., which we call 
"physical quantities". It is now recognised as essential 
that these should be defined according to the way in 
which we actually recognise them when confronted with 
them, and not according to the metaphysical significance 
which we may have anticipated for them. In the old 
textbooks mass was defined as "quantity of matter"; 



NATURE OF EXACT SCIENCE 255 

but when it came to an actual determination of mass, an 
experimental method was prescribed which had no 
bearing on this definition. The belief that the quantity 
determined by the accepted method of measurement 
represented the quantity of matter in the object was 
merely a pious opinion. At the present day there is no 
sense in which the quantity of matter in a pound of lead 
can be said to be equal to the quantity in a pound of 
sugar. Einstein's theory makes a clean sweep of these 
pious opinions, and insists that each physical quantity 
should be defined as the result of certain operations of 
measurement and calculation. You may if you like 
think of mass as something of inscrutable nature to 
which the pointer reading has a kind of relevance. But 
in physics at least there is nothing much to be gained 
by this mystification, because it is the pointer reading 
itself which is handled in exact science; and if you 
embed it in something of a more transcendental nature, 
you have only the extra trouble of digging it out 
again. 

It is quite true that when we say the mass is two tons 
we have not specially in mind the reading of the particu- 
lar machine on which the weighing was carried out. 
That is because we do not start to tackle the problem of 
the elephant's escapade ab initio as though it were the 
first inquiry we had ever made into the phenomena of 
the external world. The examiner would have had to be 
much more explicit if he had not presumed a general 
acquaintance with the elementary laws of physics, i.e. 
laws which permit us to deduce the readings of other 
indicators from the reading of one. // is this connec- 
tivity of pointer readings, expressed by physical laws, 
which supplies the continuous background that any realis- 
tic problem demands. 



256 POINTER READINGS 

It is obviously one of the conditions of the problem 
that the same elephant should be concerned in the 
weighing experiment and in the tobogganing experi- 
ment. How can this identity be expressed in a descrip- 
tion of the world by pointer readings only? Two 
readings may be equal, but it is meaningless to inquire 
if they are identical; if then the elephant is a bundle of 
pointer readings, how can we ask whether it is continu- 
ally the identical bundle ? The examiner does not confide 
to us how the identity of the elephant was ensured; we 
have only his personal guarantee that there was no 
substitution. Perhaps the creature answered to its name 
on both occasions; if so the test of identity is clearly 
outside the present domain of physics. The only test 
lying purely in the domain of physics is that of con- 
tinuity; the elephant must be watched all the way from 
the scales to the hillside. The elephant, we must remem- 
ber, is a tube in the four-dimensional world demarcated 
from the rest of space-time by a more or less abrupt 
boundary. Using the retina of his eye as an indicator 
and making frequent readings of the oudine of the 
image, the observer satisfied himself that he was fol- 
lowing one continuous and isolated world-tube from 
beginning to end. If his vigilance was intermittent he 
took a risk of substitution, and consequently a risk of 
the observed time of descent failing to agree with the 
time calculated.* Note that we do not infer that there 
is any identity of the contents of the isolated world-tube 
throughout its length; such identity would be meaning- 

* A good illustration of such substitution is afforded by astronomical 
observations of a certain double star with two components of equal 
brightness. After an intermission of observation the two components 
were inadvertently interchanged, and the substitution was not detected 
until the increasing discrepancy between the actual and predicted orbits 
was inquired into. 



LIMITS OF PHYSICAL KNOWLEDGE 257 

less in physics. We use instead the law of conserva- 
tion of mass (either as an empirical law or deduced 
from the law of gravitation) which assures us that, 
provided the tube is isolated, the pointer reading on 
the schedule derived from the weighing-machine type 
of experiment has a constant value along the tube. 
For the purpose of exact science "the same object" 
becomes replaced by "isolated world-tube". The con- 
stancy of certain properties of the elephant is not 
assumed as self-evident from its sameness, but is an 
inference from experimental and theoretical laws re- 
lating to world-tubes which are accepted as well 
established. 

Limitations of Physical Knowledge, Whenever we state 
the properties of a body in terms of physical quantities 
we are imparting knowledge as to the response of 
various metrical indicators to its presence, and nothing 
more. After all, knowledge of this kind is fairly com- 
prehensive. A knowledge of the response of all kinds 
of objects — weighing-machines and other indicators — 
would determine completely its relation to its environ- 
ment, leaving only its inner un-get-atable nature un- 
determined. In the relativity theory we accept this as 
full knowledge, the nature of an object in so far as it is 
ascertainable by scientific inquiry being the abstraction 
of its relations to all surrounding objects. The progress 
of the relativity theory has been largely due to the 
development of a powerful mathematical calculus for 
dealing compendiously with an infinite scheme of 
pointer readings, and the technical term tensor used so 
largely in treatises on Einstein's theory may be translated 
schedule of pointer readings. It is part of the aesthetic 
appeal of the mathematical theory of relativity that the 



258 POINTER READINGS 

mathematics is so closely adapted to the physical con- 
ceptions. It is not so in all subjects. For example, we 
may admire the triumph of patience of the mathemati- 
cian in predicting so closely the positions of the moon, 
but aesthetically the lunar theory is atrocious; it is 
obvious that the moon and the mathematician use dif- 
ferent methods of finding the lunar orbit. But by the 
use of tensors the mathematical physicist precisely de- 
scribes the nature of his subject-matter as a schedule of 
indicator readings; and those accretions of images and 
conceptions which have no place in physical science are 
automatically dismissed. 

The recognition that our knowledge of the objects 
treated in physics consists solely of readings of pointers 
and other indicators transforms our view of the status 
of physical knowledge in a fundamental way. Until 
recently it was taken for granted that we had knowledge 
of a much more intimate kind of the entities of the ex- 
ternal world. Let me give an illustration which takes 
us to the root of the great problem of the relations 
of matter and spirit. Take the living human brain 
endowed with mind and thought. Thought is one of the 
indisputable facts of the world. I know that I think, 
with a certainty which I cannot attribute to any of my 
physical knowledge of the world. More hypothetically, 
but pn fairly plausible evidence, I am convinced that 
you have minds which think. Here then is a world fact 
to be investigated. The physicist brings his tools and 
commences systematic exploration. All that he dis- 
covers is a collection of atoms and electrons and fields of 
force arranged in space and time, apparently similar to 
those found in inorganic objects. He may trace other 
physical characteristics, energy, temperature, entropy. 
None of these is identical with thought. He might set 



LIMITS OF PHYSICAL KNOWLEDGE 259 

down thought as an illusion — some perverse interpreta- 
tion of the interplay of the physical entities that he has 
found. Or if he sees the folly of calling the most un- 
doubted element of our experience an illusion, he will 
have to face the tremendous question, How can this col- 
lection of ordinary atoms be a thinking machine? But 
what knowledge have we of the nature of atoms which 
renders it at all incongruous that they should constitute 
a thinking object? The Victorian physicist felt that he 
knew just what he was talking about when he used such 
terms as matter and atoms. Atoms were tiny billiard 
balls, a crisp statement that was supposed to tell you 
all about their nature in a way which could never be 
achieved for transcendental things like consciousness, 
beauty or humour. But now we realise that science has 
nothing to say as to the intrinsic nature of the atom. The 
physical atom is, like everything else in physics, a 
schedule of pointer readings. The schedule is, we agree, 
attached to some unknown background. Why not then 
attach it to something of spiritual nature of which a 
prominent characteristic is thought. It seems rather silly 
to prefer to attach it to something of a so-called "con- 
crete" nature inconsistent with thought, and then to 
wonder where the thought comes from. We have dis- 
missed all preconception as to the background of our 
pointer readings, and for the most part we can discover 
nothing as to its nature. But in one case — namely, for 
the pointer readings of my own brain — I have an in- 
sight which is not limited to the evidence of the pointer 
readings. That insight shows that they are attached to 
a background of consciousness. Although I may expect 
that the background of other pointer readings in physics 
is of a nature continuous with that revealed to me in this 
particular case, I do not suppose that it always has the 



260 POINTER READINGS 

more specialised attributes of consciousness.* But in 
regard to my one piece of insight into the background 
no problem of irreconcilability arises; I have no other 
knowledge of the background with which to reconcile it. 
In science we study the linkage of pointer readings 
with pointer readings. The terms link together in endless 
cycle with the same inscrutable nature running through 
the whole. There is nothing to prevent the assemblage 
of atoms constituting a brain from being of itself a 
thinking object in virtue of that nature which physics 
leaves undetermined and undeterminable. If we must 
embed our schedule of indicator readings in some kind 
of background, at least let us accept the only hint we 
have received as to the significance of the background — 
namely that it has a nature capable of manifesting itself 
as mental activity. 

Cyclic Method of Physics. I must explain this reference 
to an endless cycle of physical terms. I will refer again 
to Einstein's law of gravitation. I have already ex- 
pounded it to you more than once and I hope you gained 
some idea of it from the explanation. This time I am 
going to expound it in a way so complete that there is 
not much likelihood that anyone will understand it. 
Never mind. We are not now seeking further light on 
the cause of gravitation; we are interested in seeing 

* For example, we should most of us assume (hypothetically) that 
the dynamical quality of the world referred to in chapter v is characteris- 
tic of the whole background. Apparently it is not to be found in the 
pointer readings, and our only insight into it is in the feeling of "becom- 
ing" in our consciousness. "Becoming" like "reasoning" is known to us 
only through its occurrence in our own minds; but whereas it would be 
absurd to suppose that the latter extends to inorganic aggregations of 
atoms, the former may be (and commonly is) extended to the inorganic 
world, so that it is not a matter of indifference whether the progress of 
the inorganic world is viewed from past to future or from future to past. 



CYCLIC METHOD OF PHYSICS 261 

what would really be involved in a complete explanation 
of anything physical. 

Einstein's law in its analytical form is a statement that 
in empty space certain quantities called potentials obey 
certain lengthy differential equations. We make a 
memorandum of the word ''potential" to remind us 
that we must later on explain what it means. We might 
conceive a world in which the potentials at every moment 
and every place had quite arbitrary values. The actual 
world is not so unlimited, the potentials being restricted 
to those values which conform to Einstein's equations. 
The next question is, What are potentials? They can 
be defined as quantities derived by quite simple mathe- 
matical calculations from certain fundamental quantities 
called intervals. (Mem. Explain "interval".) If we 
know the values of the various intervals throughout the 
world definite rules can be given for deriving the values 
of the potentials. What are intervals? They are rela- 
tions between pairs of events which can be measured with 
a scale or a clock or with both. (Mem. Explain "scale" 
and "clock".) Instructions can be given for the correct 
use of the scale and clock so that the interval is given by 
a prescribed combination of their readings. What are 
scales and clocks? A scale is a graduated strip of mat- 
ter which. . . . (Mem. Explain "matter".) On second 
thoughts I will leave the rest of the description as "an 
exercise to the reader" since \t would take rather a long 
time to enumerate all the properties and niceties of 
behaviour of the material standard which a physicist 
would accept as a perfect scale or a perfect clock. We 
pass on to the next question, What is matter? We have 
dismissed the metaphysical conception of substance. We 
might perhaps here describe the atomic and electrical 
structure of matter, but that leads to the microscopic 



262 POINTER READINGS 

aspects of the world, whereas we are here taking the 
macroscopic outlook. Confining ourselves to mechanics, 
which is the subject in which the law of gravitation 
arises, matter may be defined as the embodiment of three 
related physical quantities, mass (or energy), momentum 
and stress. What are "mass", "momentum" and 
"stress"? It is one of the most far-reaching achieve- 
ments of Einstein's theory that it has given an exact 
answer to this question. They are rather formidable 
looking expressions containing the potentials and their 
first and second derivatives with respect to the co- 
ordinates. What are the potentials? Why, that is just 
what I have been explaining to you! 

The definitions of physics proceed according to the 
method immortalised in "The House that Jack built" : 
This is the potential, that was derived from the interval, 
that was measured by the scale, that was made from the 
matter, that embodied the stress, that. . . . But instead 
of finishing with Jack, whom of course every youngster 
must know without need for an introduction, we make 
a circuit back to the beginning of the rhyme: . . . that 
worried the cat, that killed the rat, that ate the malt, 
that lay in the house, that was built by the priest all 
shaven and shorn, that married the man. . . . Now we 
can go round and round for ever. 

But perhaps you have already cut short my explana- 
tion of gravitation. When we reached matter you had 
had enough of it. "Please do not explain any more, 
I happen to know what matter is." Very well; matter 
is something that Mr. X knows. Let us see how it goes : 
This is the potential that was derived from the interval 
that was measured by the scale that was made from the 
matter that Mr. X knows. Next question, What is Mr. X? 

Well, it happens that physics is not at all anxious to 



CYCLIC METHOD OF PHYSICS 263 

pursue the question, What is Mr. X? It is not disposed 
to admit that its elaborate structure of a physical uni- 
verse is ''The House that Mr. X built". It looks upon 
Mr. X — and more particularly the part of Mr. X that 
knows — as a rather troublesome tenant who at a late 
stage of the world's history has come to inhabit a 

Potential 




Stress m •Interval 




Matter • < • Scale 



Mr. X • 




Fig. 8 

structure which inorganic Nature has by slow evolutionary 
progress contrived to build. And so it turns aside from 
the avenue leading to Mr. X — and beyond — and closes 
up its cycle leaving him out in the cold. 

From its own point of view physics is entirely jus- 
tified. That matter in some indirect way comes within 
the purview of Mr. X's mind is not a fact of any utility 



264 POINTER READINGS 

for a theoretical scheme of physics. We cannot embody 
it in a differential equation. It is ignored; and the 
physical properties of matter and other entities are 
expressed by their linkages in the cycle. And you can 
see how by the ingenious device of the cycle physics 
secures for itself a self-contained domain for study with 
no loose ends projecting into the unknown. All other 
physical definitions have the same kind of interlocking. 
Electric force is defined as something which causes 
motion of an electric charge ; an electric charge is some- 
thing which exerts electric force. So that an electric 
charge is something that exerts something that produces 
motion of something that exerts something that produces 
. . . ad infinitum. 

But I am not now writing of pure physics, and from 
a broader standpoint I do not see how we can leave out 
Mr. X. The fact that matter is "knowable to Mr. X" 
must be set down as one of the fundamental attributes 
of matter. I do not say that it is very distinctive, since 
other entities of physics are also knowable to him; but 
the potentiality of the whole physical world for awaking 
impressions in consciousness is an attribute not to be 
ignored when we compare the actual world with worlds 
which, we fancy, might have been created. There seems 
to be a prevalent disposition to minimise the importance 
of this. The attitude is that "knowableness to Mr. X" 
is a negligible attribute, because Mr. X is so clever that 
he could know pretty much anything that there was to 
know. I have already urged the contrary view — that 
there is a definitely selective action of the mind; and 
since physics treats of what is knowable to mind * its 

* This is obviously true of all experimental physics, and must be 
true of theoretical physics if it is (as it professes to be) based on experi- 
ment. 



ACTUALITY 265 

subject-matter has undergone, and indeed retains evi- 
dences of, this process of selection. 

Actuality. "Knowableness to mind" is moreover a 
property which differentiates the actual world of our 
experience from imaginary worlds in which the same 
general laws of Nature are supposed to hold true. 
Consider a world — Utopia, let us say — governed by all 
the laws of Nature known and unknown which govern 
our own world, but containing better stars, planets, 
cities, animals, etc. — a world which might exist, but 
it just happens that it doesn't. How can the physicist 
test that Utopia is not the actual world? We refer to 
a piece of matter in it; it is not real matter but it attracts 
any other piece of (unreal) matter in Utopia according 
to the law of gravitation. Scales and clocks constructed 
of this unreal matter will measure wrong intervals, but 
the physicist cannot detect that they are wrong unless 
he has first shown the unreality of the matter. As soon 
as any element in it has been shown to be unreal Utopia 
collapses; but so long as we keep to the cycles of physics 
we can never find the vulnerable point, for each element 
is correctly linked to the rest of the cycle, all our laws 
of Nature expressed by the cycle being obeyed in Utopia 
by hypothesis. The unreal stars emit unreal light which 
falls on unreal retinas and ultimately reaches unreal 
brains. The next step takes it outside the cycle and gives 
the opportunity of exposing the whole deception. Is 
the brain disturbance translated into consciousness? 
That will test whether the brain is real or unreal. There 
is no question about consciousness being real or not; 
consciousness is self-knowing and the epithet real adds 
nothing to that. Of the infinite number of worlds 
which are examples of what might be possible under the 



266 POINTER READINGS 

laws of Nature, there is one which does something more 
than fulfil those laws of Nature. This property, which 
is evidently not definable with respect to any of the laws 
of Nature, we describe as "actuality" — generally using 
the word as a kind of halo of indefinite import. We 
have seen that the trend of modern physics is to reject 
these indefinite attributions and to define its terms 
according to the way in which we recognise the pro- 
perties w T hen confronted by them. We recognise the 
actuality of a particular world because it is that world 
alone with which consciousness interacts. However 
much the theoretical physicist may dislike a reference to 
consciousness, the experimental physicist uses freely 
this touchstone of actuality. He would perhaps prefer 
to believe that his instruments and observations are certi- 
fied as actual by his material sense organs; but the final 
guarantor is the mind that comes to know the indications 
of the material organs. Each of us is armed with this 
touchstone of actuality; by applying it we decide that 
this sorry world of ours is actual and Utopia is a dream. 
As our individual consciousnesses are different, so our 
touchstones are different; but fortunately they all agree 
in their indication of actuality — or at any rate those 
which agree are in sufficient majority to shut the others 
up in lunatic asylums. 

It is natural that theoretical physics in its formulation 
of a general scheme of law should leave out of account 
actuality and the guarantor of actuality. For it is just 
this omission which makes the difference between a law 
of Nature and a particular sequence of events. That 
which is possible (or not "too improbable") is the 
domain of natural science; that which is actual is the 
domain of natural history. We need scarcely add that 
the contemplation in natural science of a wider domain 



ACTUALITY 267 

than the actual leads to a far better understanding of 
the actual. 

From a broader point of view than that of elaborating 
the physical scheme of law we cannot treat the connection 
with mind as merely an incident in a self-existent inor- 
ganic world. In saying that the differentiation of the 
actual from the non-actual is only expressible by reference 
to mind I do not mean to imply that a universe without 
conscious mind would have no more status than Utopia. 
But its property of actuality would be indefinable since 
the one approach to a definition is cut off. The actuality 
of Nature is like the beauty of Nature. We can scarcely 
describe the beauty of a landscape as non-existent when 
there is no conscious being to witness it; but it is through 
consciousness that we can attribute a meaning to it. 
And so it is with the actuality of the world. If actuality 
means "known to mind" then it is a purely subjective 
character of the world; to make it objective we must 
substitute "knowable to mind". The less stress we lay 
on the accident of parts of the world being known at 
the present era to particular minds, the more stress we 
must lay on the potentiality of being known to mind as 
a fundamental objective property of matter, giving it 
the status of actuality whether individual consciousness 
is taking note of it or not. 

In the diagram Mr. X has been linked to the cycle at 
a particular point in deference to his supposed claim 
that he knows matter; but a little reflection will show 
that the point of contact of mind with the physical 
universe is not very definite. Mr. X knows a table; but 
the point of contact with his mind is not in the material 
of the table. Light waves are propagated from the 
table to the eye; chemical changes occur in the retina; 
propagation of some kind occurs in the optic nerves; 



268 POINTER READINGS 

atomic changes follow in the brain. Just where the 
final leap into consciousness occurs is not clear. We do 
not know the last stage of the message in the physical 
world before it became a sensation in consciousness. 
This makes no difference. The physical entities have 
a cyclic connection, and whatever intrinsic nature we 
attribute to one of them runs as a background through 
the whole cycle. It is not a question whether matter or 
electricity or potential is the direct stimulus to the mind; 
in their physical aspects these are equally represented 
as pointer readings or schedules of pointer readings. 
According to our discussion of world building they are 
the measures of structure arising from the comparability 
of certain aspects of the basal relations — measures which 
by no means exhaust the significance of those relations. 
I do not believe that the activity of matter at a certain 
point of the brain stimulates an activity of mind; my 
view is that in the activity of matter there is a metrical 
description of certain aspects of the activity of mind. 
The activity of the matter is our way of recognising a 
combination of the measures of structure; the activity 
of the mind is our insight into the complex of relations 
whose comparability gives the foundation of those 
measures. 

"What is Mr. X?" In the light of these considerations 
let us now see what we can make of the question, What 
is Mr. X? I must undertake the inquiry single-handed; 
I cannot avail myself of your collaboration without first 
answering or assuming an answer to the equally difficult 
question, What are you? Accordingly the whole in- 
quiry must take place in the domain of my own con- 
sciousness. I find there certain data purporting to 
relate to this unknown X; and I can (by using powers 



"WHAT IS MR. X?" 269 

which respond to my volition) extend the data, i.e. I can 
perform experiments on X. For example I can make 
a chemical analysis. The immediate result of these 
experiments is the occurrence of certain visual or 
olfactory sensations in my consciousness. Clearly it is 
a long stride from these sensations to any rational in- 
ference about Mr. X. For example, I learn that Mr. X 
has carbon in his brain, but the immediate knowledge 
was of something (not carbon) in my own mind. The 
reason why I, on becoming aware of something in my 
mind, can proceed to assert knowledge of something 
elsewhere, is because there is a systematic scheme of 
inference which can be traced from the one item of 
knowledge to the other. Leaving aside instinctive or 
commonsense inference — the crude precursor of scien- 
tific inference — the inference follows a linkage, which 
can only be described symbolically, extending from the 
point in the symbolic world where I locate myself to the 
point where I locate Mr. X. 

One feature of this inference is that I never discover 
what carbon really is. It remains a symbol. There is 
carbon in my own brain-mind; but the self-knowledge 
of my mind does not reveal this to me. I can only know 
that the symbol for carbon must be placed there by 
following a route of inference through the external 
world similar to that used in discovering it in Mr. X; and 
however closely associated this carbon may be with my 
thinking powers, it is as a symbol divorced from any 
thinking capacity that I learn of its existence. Carbon 
is a symbol definable only in terms of the other symbols 
belonging to the cyclic scheme of physics. What I have 
discovered is that, in order that the symbols describing 
the physical world may conform to the mathematical 
formulae which they are designed to obey, it is necessary 



270 POINTER READINGS 

to place the symbol for carbon (amongst others) in the 
locality of Mr. X. By similar means I can make an 
exhaustive physical examination of Mr. X and discover 
the whole array of symbols to be assigned to his 
locality. 

Will this array of symbols give me the whole of 
Mr. X? There is not the least reason to think so. The 
voice that comes to us over the telephone wire is not the 
whole of what is at the end of the wire. The scientific 
linkage is like the telephone wire; it can transmit just 
what it is constructed to transmit and no more. 

It will be seen that the line of communication has 
two aspects. It is a chain of inference stretching from 
the symbols immediately associated with the sensations 
in my mind to the symbols descriptive of Mr. X; and 
it is a chain of stimuli in the external world starting 
from Mr. X and reaching my brain. Ideally the steps 
of the inference exactly reverse the steps of the physical 
transmission which brought the information. (Naturally 
we make many short cuts in inference by applying 
accumulated experience and knowledge.) Commonly 
we think of it only in its second aspect as a physical 
transmission; but because it is also a line of inference 
it is subject to limitations which we should not necessarily 
expect a physical transmission to conform to. 

The system of inference employed in physical in- 
vestigation reduces to mathematical equations governing 
the symbols, and so long as we adhere to this procedure 
we are limited to symbols of arithmetical character 
appropriate to such mathematical equations.* Thus 
there is no opportunity for acquiring by any physical 

* The solitary exception is, I believe, Dirac's generalisation which 
introduces g-numbers (p. 210). There is as yet no approach to a general 
system of inference on a non-numerical basis. 



"WHAT IS MR. X?" 271 

investigation a knowledge of Mr. X other than that 
which can be expressed in numerical form so as to 
be passed through a succession of mathematical 
equations. 

Mathematics is the model of exact inference; and 
in physics we have endeavoured to replace all cruder 
inference by this rigorous type. Where we cannot 
complete the mathematical chain we confess that we are 
wandering in the dark and are unable to assert real 
knowledge. Small wonder then that physical science 
should have evolved a conception of the world consisting 
of entities rigorously bound to one another by mathe- 
matical equations forming a deterministic scheme. This 
knowledge has all been inferred and it was bound there- 
fore to conform to the system of inference that was used. 
The determinism of the physical laws simply reflects 
the determinism of the method of inference. This soulless 
nature of the scientific world need not worry those who 
are persuaded that the main significances of our en- 
vironment are of a more spiritual character. Anyone 
who studied the method of inference employed by the 
physicist could predict the general characteristics of 
the world that he must necessarily find. What he could 
not have predicted is the great success of the method — 
the submission of so large a proportion of natural 
phenomena to be brought into the prejudged scheme. 
But making all allowance for future progress in develop- 
ing the scheme, it seems to be flying in the face of 
obvious facts to pretend that it is all comprehensive, 
Mr. X is one of the recalcitrants. When sound-waves 
impinge on his ear he moves, not in accordance with a 
mathematical equation involving the physical measure 
numbers of the waves, but in accordance with the 
meaning that those sound-waves are used to convey. To 



272 POINTER READINGS 

know what there is about Mr. X which makes him 
behave in this strange way, we must look not to a 
physical system of inference, but to that insight beneath 
the symbols which in our own minds we possess. It is 
by this insight that we can finally reach an answer to 
our question, What is Mr. X? 



Chapter XIII 

REALITY 

The Real and the Concrete. One of our ancestors, taking 
arboreal exercise in the forest, failed to reach the bough 
intended and his hand closed on nothingness. The 
accident might well occasion philosophical reflections 
on the distinctions of substance and void — to say nothing 
of the phenomenon of gravity. However that may be, 
his descendants down to this day have come to be 
endowed with an immense respect for substance arising 
we know not how or why. So far as familiar experience 
is concerned, substance occupies the centre of the stage, 
rigged out with the attributes of form, colour, hardness, 
etc., which appeal to our several senses. Behind it is a 
subordinate background of space and time permeated 
by forces and unconcrete agencies to minister to the 
star performer. 

Our conception of substance is only vivid so long as 
we do not face it. It begins to fade when we analyse it. 
We may dismiss many of its supposed attributes which 
are evidently projections of our sense-impressions out- 
wards into the external world. Thus the colour which is 
so vivid to us is in our minds and cannot be embodied 
in a legitimate conception of the substantial object itself. 
But in any case colour is no part of the essential nature 
of substance. Its supposed nature is that which we try 
to call to mind by the word "concrete", which is 
perhaps an outward projection of our sense of touch. 

273 



274 REALITY 

When I try to abstract from the bough everything but 
its substance or concreteness and concentrate on an 
effort to apprehend this, all ideas elude me; but the 
effort brings with it an instinctive tightening of the 
fingers — from which perhaps I might infer that my 
conception of substance is not very different from my 
arboreal ancestor's. 

So strongly has substance held the place of leading 
actor on the stage of experience that in common usage 
concrete and real are almost synonymous. Ask any man 
who is not a philosopher or a mystic to name something 
typically real; he is almost sure to choose a concrete 
thing. Put the question to him whether Time is real; 
he will probably decide with some hesitation that it 
must be classed as real, but he has an inner feeling that 
the question is in some way inappropriate and that he is 
being cross-examined unfairly. 

In the scientific world the conception of substance 
is wholly lacking, and that which most nearly replaces 
it, viz. electric charge, is not exalted as star-performer 
above the other entities of physics. For this reason the 
scientific world often shocks us by its appearance of 
unreality. It offers nothing to satisfy our demand for 
the concrete. How should it, when we cannot formu- 
late that demand? I tried to formulate it; but nothing 
resulted save a tightening of the fingers. Science does 
not overlook the provision for tactual and muscular 
sensation. In leading us away from the concrete, science 
is reminding us that our contact with the real is more 
varied than was apparent to the ape-mind, to whom the 
bough which supported him typified the beginning and 
end of reality. 

It is not solely the scientific world that will now 
occupy our attention. In accordance with the last 



THE REAL AND THE CONCRETE 275 

chapter we are takfng a larger view in which the cyclical 
schemes of physics are embraced with much besides. 
But before venturing on this more risky ground I have 
to emphasise one conclusion which is definitely scien- 
tific. The modern scientific theories have broken away 
from the common standpoint which identifies the real 
with the concrete. I think we might go so far as to say 
that time is more typical of physical reality than matter, 
because it is freer from those metaphysical associations 
which physics disallows. It would not be fair, being 
given an inch, to take an ell, and say that having gone 
so far physics may as well admit at once that reality is 
spiritual. We must go more warily. But in approaching 
such questions we are no longer tempted to take up the 
attitude that everything which lacks concreteness is 
thereby self-condemned. 

The cleavage between the scientific and the extra- 
scientific domain of experience is, I believe, not a 
cleavage between the concrete and the transcendental 
but between the metrical and the non-metrical. I am 
at one with the materialist in feeling a repugnance 
towards any kind of pseudo-science of the extra- 
scientific territory. Science is not to be condemned as 
narrow because it refuses to deal with elements of 
experience which are unadapted to its own highly 
organised method ; nor can it be blamed for looking super- 
ciliously on the comparative disorganisation of our knowl- 
edge and methods of reasoning about the non-metrical 
part of experience. But I think we have not been guilty 
of pseudo-science in our attempt to show in the last two 
chapters how it comes about that within the whole 
domain of experience a selected portion is capable of 
that exact metrical representation which is requisite for 
development by the scientific method. 



276 REALITY 

Mind-Stuff. I will try to be as definite as I can as to the 
glimpse of reality which we seem to have reached. Only 
I am well aware that in committing myself to details 
I shall probably blunder. Even if the right view has 
here been taken of the philosophical trend of modern 
science, it is premature to suggest a cut-and-dried 
scheme of the nature of things. If the criticism is made 
that certain aspects are touched on which come more 
within the province of the expert psychologist, I must 
admit its pertinence. The recent tendencies of science 
do, I believe, take us to an eminence from which we 
can look down into the deep waters of philosophy; and 
if I rashly plunge into them,, it is not because I have 
confidence in my powers of swimming, but to try to 
show that the water is really deep. 

To put the conclusion crudely — the stuff of the world 
is mind-stuff. As is often the way with crude statements, 
I shall have to explain that by "mind" I do not here 
exactly mean mind and by "stuff" I do not at all mean 
stuff. Still this is about as near as we can get to the idea 
in a simple phrase. The mind-stuff of the world is, of 
course, something more general than our individual 
conscious minds; but we may think of its nature as not 
altogether foreign to the feelings in our consciousness. 
The realistic matter and fields of force of former 
physical theory are altogether irrelevant — except in so 
far as the mind-stuff has itself spun these imaginings. 
The symbolic matter and fields of force of present-day 
theory are more relevant, but they bear to it the same 
relation that the bursar's accounts bear to the activity 
of the college. Having granted this, the mental activity 
of the part of the world constituting ourselves occasions 
no surprise; it is known to us by direct self-knowledge, 
and we do not explain it away as something other than 



MIND-STUFF 277 

we know it to be — or, rather, it knows itself to be. It 
is the physical aspects of the world that we have to 
explain, presumably by some such method as that set 
forth in our discussion on world-building. Our bodies 
are more mysterious than our minds — at least they 
would be, only that we can set the mystery on one side 
by the device of the cyclic scheme of physics, which 
enables us to study their phenomenal behaviour without 
ever coming to grips with the underlying mystery. 

The mind-stuff is not spread in space and time; these 
are part of the cyclic scheme ultimately derived out of 
it. But we must presume that in some other way or 
aspect it can be differentiated into parts. Only here and 
there does it rise to the level of consciousness, but from 
such islands proceeds all knowledge. Besides the direct 
knowledge contained in each self-knowing unit, there 
is inferential knowledge. The latter includes our know- 
ledge of the physical world. It is necessary to keep 
reminding ourselves that all knowledge of our environ- 
ment from which the world of physics is constructed, 
has entered in the form of messages transmitted along 
the nerves to the seat of consciousness. Obviously the 
messages travel in code. When messages relating to a 
table are travelling in the nerves, the nerve-disturbance 
does not in the least resemble either the external table 
that originates the mental impression or the conception 
of the table that arises in consciousness.* In the central 
clearing station the incoming messages are sorted and 
decoded, partly by instinctive image-building inherited 

*I mean, resemble in intrinsic nature. It is true (as Bertrand Russell 
has emphasised) that the symbolic description of structure will be iden- 
tical for the t table in the external world and for the conception of the 
table in consciousness if the conception is scientifically correct. If the 
physicist does not attempt to penetrate beneath the structure he is in- 
different as to which of the two we imagine ourselves to be discussing. 



278 REALITY 

from the experience of our ancestors, partly by scientific 
comparison and reasoning. By this very indirect and 
hypothetical inference all our supposed acquaintance 
with and our theories of a world outside us have been 
built up. We are acquainted with an external world 
because its fibres run into our consciousness; it is only 
our own ends of the fibres that we actually know; from 
those ends we more or less successfully reconstruct the 
rest, as a palaeontologist reconstructs an extinct monster 
from its footprint. 

The mind-stuff is the aggregation of relations and 
relata which form the building material for the physical 
world. Our account of the building process shows, 
however, that much that is implied in the relations is 
dropped as unserviceable for the required building. 
Our view is practically that urged in 1875 by W. K. 
Clifford— 

"The succession of feelings which constitutes a man's 
consciousness is the reality which produces in our minds 
the perception of the motions of his brain." 

That is to say, that which the man himself knows as 
a succession of feelings is the reality which when probed 
by the appliances of an outside investigator affects their 
readings in such a way that it is identified as a configura- 
tion of brain-matter. Again Bertrand Russell writes — * 

What the physiologist sees when he examines a brain is in the 
physiologist, not in the brain he is examining. What is in the 
brain by the time the physiologist examines it if it is dead, I do 
not profess to know; but while its owner was alive, part, at least, 
of the contents of his brain consisted of his percepts, thoughts, 
and feelings. Since his brain also consisted of electrons, we are 
compelled to conclude that an electron is a grouping of events, 

* Analysis of Matter, p. 320. 



MIND-STUFF 279 

and that if the electron is in a human brain, some of the events 
composing it are likely to be some of the "mental states" of the 
man to whom the brain belongs. Or, at any rate, they are likely 
to be parts of such "mental states" — for it must not be assumed 
that part of a mental state must be a mental state. I do not wish 
to discuss what is meant by a "mental state"; the main point for 
us is that the term must include percepts. Thus a percept is an 
event or a group of events, each of which belongs to one or more 
of the groups constituting the electrons in the brain. This, 
I think, is the most concrete statement that can be made about 
electrons; everything else that can be said is more or less abstract 
and mathematical. 

I quote this partly for the sake of the remark that it 
must not be assumed that part of a mental state must 
necessarily be a mental state. We can no doubt analyse 
the content of consciousness during a short interval of 
time into more or less elementary constituent feelings; 
but it is not suggested that this psychological analysis 
will reveal the elements out of whose measure-numbers 
the atoms or electrons are built. The brain-matter is a 
partial aspect of the whole mental state; but the analysis 
of the brain-matter by physical investigation does not 
run at all parallel with the analysis of the mental state 
by psychological investigation. I assume that Russell 
meant to warn us that, in speaking of part of a mental 
state, he was not limiting himself to parts that would 
be recognised as such psychologically, and he was ad- 
mitting a more abstract kind of dissection. 

This might give rise to some difficulty if we were 
postulating complete identity of mind-stuff with con- 
sciousness. But we know that in the mind there are 
memories not in consciousness at the moment but 
capable of being summoned into consciousness. We 
are vaguely aware that things we cannot recall are lying 
somewhere about and may come into the mind at any 



280 REALITY 

moment. Consciousness is not sharply defined, but 
fades into subconsciousness; and beyond that we must 
postulate something indefinite but yet continuous with 
our mental nature. This I take to be the world-stuff. 
We liken it to our conscious feelings because, now that 
we are convinced of the formal and symbolic character of 
the entities of physics, there is nothing else to liken it to. 

It is sometimes urged that the basal stuff of the world 
should be called "neutral stuff" rather than "mind- 
stuff", since it is to be such that both mind and matter 
originate from it. If this is intended to emphasise that 
only limited islands of it constitute actual minds, and 
that even in these islands that which is known mentally 
is not equivalent to a complete inventory of all that may 
be there, I agree. In fact I should suppose that the 
self-knowledge of consciousness is mainly or wholly a 
knowledge which eludes the inventory method of de- 
scription. The term "mind-stuff" might well be amended; 
but neutral stuff seems to be the wrong kind of amend- 
ment. It implies that we have two avenues of approach 
to an understanding of its nature. We have only one 
approach, namely, through our direct knowledge of 
mind. The supposed approach through the physical 
world leads only into the cycle of physics, where we run 
round and round like a kitten chasing its tail and never 
reach the world-stuff at all. 

I assume that we have left the illusion of substance 
so far behind that the word "stuff" will not cause any 
misapprehension. I certainly do not intend to materialise 
or substantialise mind. Mind is — but you know what 
mind is like, so why should I say more about its nature? 
The word "stuff" has reference to the function it has 
to perform as a basis of world-building and does not 
imply any modified view of its nature. 



MIND-STUFF 281 

It is difficult for the matter-of-fact physicist to accept 
the view that the substratum of everything is of mental 
character. But no one can deny that mind is the first 
and most direct thing in our experience, and all else is 
remote inference — inference either intuitive or deli- 
berate. Probably it would never have occurred to us 
(as a serious hypothesis) that the world could be based 
on anything else, had we not been under the impression 
that there was a rival stuff with a more comfortable kind 
of "concrete" reality — something too inert and stupid 
to be capable of forging an illusion. The rival turns 
out to be a schedule of pointer readings; and though a 
world of symbolic character can well be constructed from 
it, this is a mere shelving of the inquiry into the nature 
of the world of experience. 

This view of the relation of the material to the 
spiritual world perhaps relieves to some extent a tension 
between science and religion. Physical science has 
seemed to occupy a domain of reality which is self- 
sufficient, pursuing its course independently of and 
indifferent to that which a voice within us asserts to be 
a higher reality. We are jealous of such independence. 
We are uneasy that there should be an apparently self- 
contained world in which God becomes an unnecessary 
hypothesis. We acknowledge that the ways of God are 
inscrutable; but is there not still in the religious mind 
something of that feeling of the prophets of old, who 
called on God to assert his kingship and by sign or 
miracle proclaim that the forces of Nature are subject 
to his command? And yet if the scientist were to repent 
and admit that it was necessary to include among 
the agents controlling the stars and the electrons an omni- 
present spirit to whom we trace the sacred things of con- 
sciousness, would there not be even graver apprehension ? 



282 REALITY 

We should suspect an intention to reduce God to a system 
of differential equations, like the other agents which at 
various times have been introduced to restore order in the 
physical scheme. That fiasco at any rate is avoided. For 
the sphere of the differential equations of physics is the 
metrical cyclic scheme extracted out of the broader 
reality. However much the ramifications of the cycles 
may be extended by further scientific discovery, they 
cannot from their very nature trench on the background 
in which they have their being — their actuality. It is 
in this background that our own mental consciousness lies; 
and here, if anywhere, we may find a Power greater than 
but akin to consciousness. It is not possible for the con- 
trolling laws of the spiritual substratum, which in so far 
as it is known to us in consciousness is essentially non- 
metrical, to be analogous to the differential and other 
mathematical equations of physics which are meaningless 
unless they are fed with metrical quantities. So that the 
crudest anthropomorphic image of a spiritual deity can 
scarcely be so wide of the truth as one conceived in terms 
of metrical equations. 

The Definition of Reality. It is time we came to grips 
with the loose terms Reality and Existence, which we 
have been using without any inquiry into what they are 
meant to convey. I am afraid of this word Reality, not 
connoting an ordinarily definable characteristic of the 
things it is applied to but used as though it were some 
kind of celestial halo. I very much doubt if any one of 
us has the faintest idea of what is meant by the reality 
or existence of anything but our own Egos. That is a 
bold statement, which I must guard against misinter- 
pretation. It is, of course, possible to obtain consistent 
use of the word "reality" by adopting a conventional 



THE DEFINITION OF REALITY 283 

definition. My own practice would probably be covered 
by the definition that a thing may be said to be real if 
it is the goal of a type of inquiry to which I personally 
attach importance. But if I insist on no more than this 
I am whittling down the significance that is generally 
assumed. In physics we can give a cold scientific 
definition of reality which is free from all sentimental 
mystification. But this is not quite fair play, because the 
word "reality" is generally used with the intention of 
evoking sentiment. It is a grand word for a peroration. 
"The right honourable speaker went on to declare that 
the concord and amity for which he had unceasingly 
striven had now become a reality (loud cheers). " The 
conception which it is so troublesome to apprehend is 
not "reality" but "reality (loud cheers)". 

Let us first examine the definition according to the 
purely scientific usage of the word, although it will not 
take us far enough. The only subject presented to me 
for study is the content of my consciousness. You are 
able to communicate to me part of the content of your 
consciousness which thereby becomes accessible in my 
own. For reasons which are generally admitted, though 
I should not like to have to prove that they are conclusive, 
I grant your consciousness equal status with my own; 
and I use this second-hand part of my consciousness to 
"put myself in your place". Accordingly my subject of 
study becomes differentiated into the contents of many 
consciousnesses, each content constituting a view-point. 
There then arises the problem of combining the view- 
points, and it is through this that the external world of 
physics arises. Much that is in any one consciousness 
is individual, much is apparently alterable by volition; 
but there is a stable element which is common to other' 
consciousnesses. That common element we desire to 



284 REALITY 

study, to describe as fully and accurately as possible, 
and to discover the laws by which it combines now with 
one view-point, now with another. This common ele- 
ment cannot be placed in one man's consciousness 
rather than in another's; it must be in neutral ground — 
an external world. 

It is true that I have a strong impression of an external 
world apart from any communication with other con- 
scious beings. But apart from such communication 
I should have no reason to trust the impression. Most 
of our common impressions of substance, world-wide 
instants, and so on, have turned out to be illusory, and 
the externality of the world might be equally untrust- 
worthy. The impression of externality is equally strong 
in the world that comes to me in dreams; the dream- 
world is less rational, but that might be used as an argu- 
ment in favour of its externality as showing its dissocia- 
tion from the internal faculty of reason. So long as we 
have to deal with one consciousness alone, the hypothesis 
that there is an external world responsible for part of 
what appears in it is an idle one. All that can be asserted 
of this external world is a mere duplication of the know- 
ledge that can be much more confidently asserted of the 
world appearing in the consciousness. The hypothesis 
only becomes useful when it is the means of bringing 
together the worlds of many consciousnesses occupying 
different view-points. 

The external world of physics is thus a symposium 
of the worlds presented to different view-points. There 
is general agreement as to the principles on which the 
symposium should be formed. Statements made about 
this external world, if they are unambiguous, must be 
either true or false. This has often been denied by 
philosophers. It is quite commonly said that scientific 



THE DEFINITION OF REALITY 285 

theories about the world are neither true nor false but 
merely convenient or inconvenient. A favourite phrase 
is that the gauge of value of a scientific theory is that it 
economises thought. Certainly a simple statement is 
preferable to a circumlocutory one; and as regards any 
current scientific theory, it is much easier to show that 
it is convenient or that it economises thought than that 
it is true. But whatever lower standards we may apply 
in practice we need not give up our ideals; and so long 
as there is a distinction between true and false theories 
our aim must be to eliminate the false. For my part 
I hold that the continual advance of science is not a 
mere utilitarian progress; it is progress towards ever 
purer truth. Only let it be understood that the truth 
we seek in science is the truth about an external world 
propounded as the theme of study, and is not bound up 
with any opinion as to the status of that world — whether 
or not it wears the halo of reality, whether or not it is 
deserving of "loud cheers". 

Assuming that the symposium has been correctly 
carried out, the external world and all that appears in it 
are called real without further ado. When we (scientists) 
assert of anything in the external world that it is real 
and that it exists, we are expressing our belief that the 
rules of the symposium have been correctly applied — 
that it is not a false concept introduced by an error in 
the process of synthesis, or a iiallucination belonging to 
only one individual consciousness, or an incomplete 
representation which embraces certain view-points but 
conflicts with others. We refuse to contemplate the 
awful contingency that the external world, after all our 
care in arriving at it, might be disqualified by failing 
to exist; because we have no idea what the supposed 
qualification would consist in, nor in what way the 



286 REALITY 

prestige of the world would be enhanced if it passed 
the implied test. The external world is the world that 
confronts that experience which we have in common, 
and for us no other world could fill the same role, no 
matter how high honours it might take in the qualifying 
examination. 

This domestic definition of existence for scientific 
purposes follows the principle now adopted for all other 
definitions in science, namely, that a thing must be 
defined according to the way in which it is in practice 
recognised and not according to some ulterior signi- 
ficance that we imagine it to possess. Just as matter 
must shed its conception of substantiality, so existence 
must shed its halo, before we can admit it into physical 
science. But clearly if we are to assert or to question 
the existence of anything not comprised in the external 
world of physics, we must look beyond the physical 
definition. The mere questioning of the reality of the 
physical world implies some higher censorship than the 
scientific method itself can supply. 

The external world of physics has been formulated 
as an answer to a particular problem encountered in 
human experience. Officially the scientist regards it as 
a problem which he just happened across, as he might 
take up a cross-word problem encountered in a news- 
paper. His sole business is to see that the problem is 
correctly solved. But questions may be raised about a 
problem which play no part and need not be considered 
in connection with the solving of the problem. The 
extraneous question naturally raised about the problem 
of the external world is whether there is some higher 
justification for embarking on this world-solving com- 
petition rather than on other problems which our 
experience might suggest to us. Just what kind of 



THE DEFINITION OF REALITY 287 

justification the scientist would claim for his quest is not 
very clear, because it is not within the province of science 
to formulate such a claim. But certainly he makes 
claims which do not rest on the aesthetic perfection of 
the solution or on material benefits derived from scien- 
tific research. He would not allow his subject to be 
shoved aside in a symposium on truth. We can scarcely 
say anything more definite than that science claims a 
"halo" for its world. 

If we are to find for the atoms and electrons of the 
external world not merely a conventional reality but 
"reality (loud cheers)" we must look not to the end but 
to the beginning of the quest. It is at the beginning that 
we must find that sanction which raises these entities 
above the mere products of an arbitrary mental exercise. 
This involves some kind of assessment of the impulse 
which sets us forth on the voyage of discovery. How 
can we make such assessment? Not by any reasoning 
that I know of. Reasoning would only tell us that the 
impulse might be judged by the success of the adventure 
— whether it leads in the end to things which really 
exist and wear the halo in their own right; it takes us 
to and fro like a shuttle along the chain of inference in 
vain search for the elusive halo. But, legitimately or not, 
the mind is confident that it can distinguish certain 
quests as sanctioned by indisputable authority. We 
may put it in different ways ;- the impulse to this quest 
is part of our very nature; it is the expression of a 
purpose which has possession of us. Is this precisely 
what we meant when we sought to affirm the reality of 
the external world? It goes some way towards giving 
it a meaning but is scarcely the full equivalent. I doubt 
if we really satisfy the conceptions behind that demand 
unless we make the bolder hypothesis that the quest 



288 REALITY 

and all that is reached by it are of worth in the eyes of 
an Absolute Valuer. 

Whatever justification at the source we accept to 
vindicate the reality of the external world, it can scarcely 
fail to admit on the same footing much that is outside 
physical science. Although no long chains of regularised 
inference depend from them we recognise that other 
fibres of our being extend in directions away from 
sense-impressions. I am not greatly concerned to borrow 
words like "existence" and "reality" to crown these 
other departments of the soul's interest. I would rather 
put it that any raising of the question of reality in its 
transcendental sense (whether the question emanates 
from the world of physics or not) leads us to a perspective 
from which we see man not as a bundle of sensory 
impressions, but conscious of purpose and responsi- 
bilities to which the external world is subordinate. 

From this perspective we recognise a spiritual world 
alongside the physical world. Experience — that is to 
say, the self cum environment — comprises more than 
can be embraced in the physical world, restricted as it 
is to a complex of metrical symbols. The physical world 
is, we have seen, the answer to one definite and urgent 
problem arising in a survey of experience; and no other 
problem has been followed up with anything like the 
same precision and elaboration. Progress towards an 
understanding of the non-sensory constituents of our 
nature is not likely to follow similar lines, and indeed 
is not animated by the same aims. If it is felt that this 
difference is so wide that the phrase spiritual world is a 
misleading analogy, I will not insist on the term. All 
I would claim is that those who in the search for truth 
start from consciousness as a seat of self-knowledge with 
interests and responsibilities not confined to the material 



PHYSICAL ILLUSTRATIONS 289 

plane, are just as much facing the hard facts of experi- 
ence as those who start from consciousness as a device 
for reading the indications of spectroscopes and micro- 
meters. 

Physical Illustrations. If the reader is unconvinced that 
there can be anything indefinite in the question whether 
a thing exists or not, let him glance at the following 
problem. Consider a distribution of matter in Einstein's 
spherical "finite but unbounded" space. Suppose that 
the matter is so arranged that every particle has an 
exactly similar particle at its antipodes. (There is some 
reason to believe that the matter would necessarily have 
this arrangement in consequence of the law of gravita- 
tion; but this is not certain.) Each group of particles 
will therefore be exactly like the antipodal group not 
only in its structure and configuration but in its entire 
surroundings; the two groups will in fact be indis- 
tinguishable by any possible experimental test. Starting 
on a journey round the spherical world we come across 
a group A, and then after going half round we come to 
an exactly similar group A' indistinguishable by any 
test; another half circle again brings us to an exactly 
similar group, which, however, we decide is the original 
group A. Now let us ponder a little. We realise that 
in any case by going on far enough we come back to the 
same group. Why do we not accept the obvious con- 
clusion that this happened when we reached A'; every- 
thing was exactly as though we had reached the starting- 
point again? We have encountered a succession of 
precisely similar phenomena but for some arbitrary 
reason have decided that only the alternate ones are 
really the same. There is no difficulty in identifying all 
of them; in that case the space is "elliptical" instead of 



29Q REALITY 

"spherical". But which is the real truth? Disregard 
the fact that I introduced A and A' to you as though 
they were not the same particles, because that begs the 
question; imagine that you have actually had this 
adventure in a world you had not been told about. You 
cannot find out the answer. Can you conceive what the 
question means ? I cannot. All that turns on the answer 
is whether we shall provide two separate haloes for A 
and A' or whether one will suffice. 

Descriptions of the phenomena of atomic physics 
have an extraordinary vividness. We see the atoms with 
their girdles of circulating electrons darting hither and 
thither, colliding and rebounding. Free electrons torn 
from the girdles hurry away a hundred times faster, 
curving sharply round the atoms with side slips and 
hairbreadth escapes. The truants are caught and 
attached to the girdles and the escaping energy shakes 
the aether into vibration. X-rays impinge on the atoms 
and toss the electrons into higher orbits. We see these 
electrons falling back again, sometimes by steps, some- 
times with a rush, caught in a cul-de-sac of metasta- 
bility, hesitating before "forbidden passages". Behind 
it all the quantum h regulates each change with mathe- 
matical precision. This is the sort of picture that appeals 
to our understanding — no insubstantial pageant to fade 
like a dream. 

The spectacle is so fascinating that we have perhaps 
forgotten that there was a time when we wanted to be 
told what an electron is. The question was never 
answered. No familiar conceptions can be woven round 
the electron; it belongs to the waiting list. Similarly 
the description of the processes must be taken with a 
grain of salt. The tossing up of the electron is a con- 
ventional way of depicting a particular change of state 



PHYSICAL ILLUSTRATIONS 291 

of the atom which cannot really be associated with 
movements in space as macroscopically conceived. 
Something unknown is doing we don't know what — that is 
what our theory amounts to. It does not sound a par- 
ticularly illuminating theory. I have read something 
like it elsewhere — 

The slithy toves 
Did gyre and gimble in the wabe. 

There is the same suggestion of activity. There is the 
same indefiniteness as to the nature of the activity and 
of what it is that is acting. And yet from so unpromising 
a beginning we really do get somewhere. We bring 
into order a host of apparently unrelated phenomena; 
we make predictions, and our predictions come off. 
The reason — the sole reason — for this progress is that 
our description is not limited to unknown agents 
executing unknown activities, but numbers are scattered 
freely in the description. To contemplate electrons 
circulating in the atom carries us no further; but by 
contemplating eight circulating electrons in one atom 
and seven circulating electrons in another we begin to 
realise the difference between oxygen and nitrogen. 
Eight slithy toves gyre and gimble in the oxygen wabe; 
seven in nitrogen. By admitting a few numbers even 
"Jabberwocky" may become scientific. We can now 
venture on a prediction; if one of its toves escapes, 
oxygen will be masquerading in a garb properly be- 
longing to nitrogen. In the stars and nebulae we do 
find such wolves in sheep's clothing which might 
otherwise have startled us. It would not be a bad 
reminder of the essential unknownness of the funda- 
mental entities of physics to translate it into "Jabber- 
wocky"; provided all numbers — all metrical attributes 



292 REALITY 

— are unchanged, it does not suffer in the least. Out 
of the numbers proceeds that harmony of natural law 
which it is the aim of science to disclose. We can grasp 
the tune but not the player. Trinculo might have been 
referring to modern physics in the words, "This is the 
tune of our catch, played by the picture of Nobody". 



Chapter XIV 

CAUSATION 

In the old conflict between freewill and predestination 
it has seemed hitherto that physics comes down heavily 
on the side of predestination. Without making ex- 
travagant claims for the scope of natural law, its moral 
sympathy has been with the view that whatever the 
future may bring forth is already foretold in the con- 
figurations of the past — 

Yea, the first Morning of Creation wrote 

What the Last Dawn of Reckoning shall read. 

I am not so rash as to invade Scotland with a solution 
of a problem which has rent her from the synod to the 
cottage. Like most other people, I suppose, I think it 
incredible that the wider scheme of Nature which 
includes life and consciousness can be completely 
predetermined; yet I have not been able to form a 
satisfactory conception of any kind of law or causal 
sequence which shall be other than deterministic. It 
seems contrary to our feeling of the dignity of the mind 
to suppose that it merely registers a dictated sequence 
of thoughts and emotions; but it seems equally con- 
trary to its dignity to put it at the mercy of impulses 
with no causal antecedents. I shall not deal with this 
dilemma. Here I have to set forth the position of 
physical science on this matter so far as it comes into 
her territory. It does come into her territory, because 
that which we call human will cannot be entirely 
dissociated from the consequent motions of the muscles 
and disturbance of the material world. On the scientific 

293 



294 CAUSATION 

side a new situation has arisen. It is a consequence of 
the advent of the quantum theory that physics is no 
longer pledged to a scheme of deterministic law. Deter- 
minism has dropped out altogether in the latest for- 
mulations of theoretical physics and it is at least open 
to doubt whether it will ever be brought back. 

The foregoing paragraph is from the manuscript of 
the original lecture delivered in Edinburgh. The attitude 
of physics at that time was one of indifference to deter- 
minism. If there existed a scheme of strictly causal law 
at the base of phenomena the search for it was not at 
present practical politics, and meanwhile another ideal 
was being pursued. The fact that a causal basis had 
been lost sight of in the new theories was fairly well 
known; many regretted it, and held that its restoration 
was imperative.* 

In rewriting this chapter a year later I have had to 
mingle with this attitude of indifference an attitude 
more definitely hostile to determinism which has arisen 
from the acceptance of the Principle of Indeterminacy 
(p. 220). There has been no time for more than a hur- 
ried examination of the far-reaching consequences of this 
principle; and I should have been reluctant to include 
"stop-press" ideas were it not that they appear to clinch 
the conception towards which the earlier developments 
were leading. The future is a combination of the causal 
influences of the past together with unpredictable ele- 
ments — unpredictable not merely because it is im- 

* A few days after the course of lectures was completed, Einstein 
wrote in his message on the Newton Centenary, "It is only in the quan- 
tum theory that Newton's differential method becomes inadequate, and 
indeed strict causality fails us. But the last word has not yet been said. 
May the spirit of Newton's method give us the power to restore unison 
between physical reality and the profoundest characteristic of Newton's 
teaching — strict causality." (Nature, 1927, March 26, p. 467.) 



CAUSATION AND TIME'S ARROW 295 

practicable to obtain the data of prediction, but because 
no data connected causally with our experience exist. 
It will be necessary to defend so remarkable a change of 
opinion at some length. Meanwhile we may note that 
science thereby withdraws its moral opposition to free- 
will. Those who maintain a deterministic theory of 
mental activity must do so as the outcome of their study 
of the mind itself and not with the idea that they are 
thereby making it more conformable with our experi- 
mental knowledge of the laws of inorganic nature. 

Causation and Time's Arrow. Cause and effect are closely 
bound up with time's arrow; the cause must precede 
the effect. The relativity of time has not obliterated this 
order. An event Here-Now can only cause events in the 
cone of absolute future; it can be caused by events in 
the cone of absolute past; it can neither cause nor be 
caused by events in the neutral wedge, since the neces- 
sary influence would in that case have to be transmitted 
with a speed faster than light. But curiously enough this 
elementary notion of cause and effect is quite incon- 
sistent with a strictly causal scheme. How can I cause 
an event in the absolute future, if the future was pre- 
determined before I was born? The notion evidently 
implies that something may be born into the world at 
the instant Here-Now, which has an influence extending 
throughout the future cone but no corresponding 
linkage to the cone of absolute past. The primary laws 
of physics do not provide for any such one-way linkage; 
any alteration in a prescribed state of the world implies 
alterations in its past state symmetrical with the altera- 
tions in its future state. Thus in primary physics, which 
knows nothing of time's arrow, there is no discrimina- 
tion of cause and effect; but events are connected by a 



296 CAUSATION 

symmetrical causal relation which is the same viewed 
from either end. 

Primary physics postulates a strictly causal scheme, 
but the causality is a symmetrical relation and not the 
one-way relation of cause and effect. Secondary physics 
can distinguish cause and effect but its foundation does 
not rest on a causal scheme and it is indifferent as to 
whether or not strict causality prevails. 

The lever in a signal box is moved and the signal 
drops. We can point out the relation of constraint 
which associates the positions of lever and signal; we 
can also find that the movements are not synchronous, 
and calculate the time-difference. But the laws of 
mechanics do not ascribe an absolute sign to this time- 
difference; so far as they are concerned we may quite 
well suppose that the drop of the signal causes the motion 
of the lever. To settle which is the cause, we have two 
options. We can appeal to the signalman who is con- 
fident that he made the mental decision to pull the lever; 
but this criterion will only be valid if we agree that there 
was a genuine decision between two possible courses 
and not a mere mental registration of what was already 
predetermined. Or we can appeal to secondary law 
which takes note of the fact that there was more of the 
random element in the world when the signal dropped 
than when the lever moved. But the feature of secon- 
dary law is that it ignores strict causation; it concerns 
itself not with what must happen but with what is 
likely to happen. Thus distinction of cause and effect 
has no meaning in the closed system of primary laws 
of physics; to get at it we have to break into the scheme, 
introducing considerations of volition or of probability 
which are foreign to it. This is rather analogous to the 
ten vanishing coefficients of curvature which could only 



CAUSATION AND TIME'S ARROW 4&) 

be recognised if the closed system of the world were 
broken into by standards foreign to it. 

For convenience I shall call the relation of effect to 
cause causation, and the symmetrical relation which does 
not distinguish between cause and effect causality. In 
primary physics causality has completely replaced 
causation. Ideally the whole world past and future is 
connected into a deterministic scheme by relations of 
causality. Up till very recently it was universally held 
that such a determinate scheme must exist (possibly 
subject to suspension by supernatural agencies outside 
the scope of physics) ; we may therefore call this the 
"orthodox" view. It was, of course, recognised that we 
were only acquainted with part of the structure of this 
causal scheme, but it was the settled aim of theoretical 
physics to discover the whole. 

This replacement in orthodox science of causation by 
causality is important in one respect. We must not let 
causality borrow an intuitive sanction which really 
belongs only to causation. We may think we have an 
intuition that the same cause cannot have two alternative 
effects; but we do not claim any intuition that the same 
effect may not spring from two alternative causes. For 
this reason the assumption of a rigid determinateness 
enforced by relations of causality cannot be said to be 
insisted on by intuition. 

What is the ground for so much ardent faith in the 
orthodox hypothesis that physical phenomena rest ulti- 
mately on a scheme of completely deterministic laws? 
I think there are two reasons — 

(i) The principal laws of Nature which have been 
discovered are apparently of this deterministic type, 
and these have furnished the great triumphs of physical 
prediction. It is natural to trust to a line of progress 



298 CAUSATION 

which has served us well in the past. Indeed it is a 
healthy attitude to assume that nothing is beyond the 
scope of scientific prediction until the limits of prediction 
actually declare themselves. 

(2) The current epistemology of science presupposes 
a deterministic scheme of this type. To modify it in- 
volves a much deeper change in our attitude to natural 
knowledge than the mere abandonment of an untenable 
hypothesis. 

In explanation of the second point we must recall 
that knowledge of the physical world has to be inferred 
from the nerve-messages which reach our brains, and 
the current epistemology assumes that there exists a 
determinate scheme of inference (lying before us 
as an ideal and gradually being unravelled). But, as has 
already been pointed out, the chains of inference are 
simply the converse of the chains of physical causality 
by which distant events are connected to the nerve- 
messages. If the scheme of transmission of these mes- 
sages through the external world is not deterministic 
then the scheme of inference as to their source cannot 
be deterministic, and our epistemology has been based 
on an impossible ideal. In that case our attitude to the 
whole scheme of natural knowledge must be profoundly 
modified. 

These reasons will be considered at length, but it is 
convenient to state here our answers to them in equally 
summary form. 

(1) In recent times some of the greatest triumphs of 
physical prediction have been furnished by admittedly 
statistical laws which do not rest on a basis of causality. 
Moreover the great laws hitherto accepted as causal 
appear on minuter examination to be of statistical 
character. 



PREDICTABILITY OF EVENTS 299 

(2) Whether or not there is a causal scheme at the 
base of atomic phenomena, modern atomic theory is not 
now attempting to find it; and it is making rapid prog- 
ress because it no longer sets this up as a practical aim. 
We are in the position of holding an epistemological 
theory of natural knowledge which does not correspond 
to actual aim of current scientific investigation. 

Predictability of Events. Let us examine a typical case 
of successful scientific prediction. A total eclipse of the 
sun visible in Cornwall is prophesied for 1 1 August 
1999. It is generally supposed that this eclipse is 
already predetermined by the present configuration of 
the sun, earth and moon. I do not wish to arouse 
unnecessary misgiving as to whether the eclipse will 
come off. I expect it will; but let us examine the grounds 
of expectation. It is predicted as a consequence of the 
law of gravitation — a law which we found in chapter vil 
to be a mere truism. That does not diminish the value 
of the prediction; but it does suggest that we may not be 
able to pose as such marvellous prophets when we come 
up against laws which are not mere truisms. I might 
venture to predict that 2 + 2 will be equal to 4 even in 
1999; but if this should prove correct it will not help 
to convince anyone that the universe (or, if you like, the 
human mind) is governed by laws of deterministic type. 
I suppose that in the most erratically governed world 
something can be predicted if truisms are not ex- 
cluded. 

But we have to look deeper than this. The law of 
gravitation is only a truism when regarded from a 
macroscopic point of view. It presupposes space, and 
measurement with gross material or optical arrange- 
ments. It cannot be refined to an accuracy beyond the 



300 CAUSATION 

limits of these gross appliances; so that it is a truism 
with a probable error — small, but not infinitely small. 
The classical laws hold good in the limit when exceed- 
ingly large quantum numbers are involved. The system 
comprising the sun, earth and moon has exceedingly 
high state-number (p. 198); and the predictability of 
its configurations is not characteristic of natural pheno- 
mena in general but of those involving great numbers 
of atoms of action — such that we are concerned not 
with individual but with average behaviour. 

Human life is proverbially uncertain; few things are 
more certain than the solvency of a life-insurance com- 
pany. The average law is so trustworthy that it may be 
considered predestined that half the children now born 
will survive the age of x years. But that does not tell us 
whether the span of life of young A. McB. is already 
written in the book of fate, or whether there is still time 
to alter it by teaching him not to run in front of motor- 
buses. The eclipse in 1999 is as safe as the balance of 
a life-insurance company; the next quantum jump of an 
atom is as uncertain as your life and mine. 

We are thus in a position to answer the main argu- 
ment for a predetermination of the future, viz. that 
observation shows the laws of Nature to be of a type 
which leads to definite predictions of the future, and it 
is reasonable to expect that any laws which remain 
undiscovered will conform to the same type. For when 
we ask what is the characteristic of the phenomena that 
have been successfully predicted, the answer is that they 
are effects depending on the average configurations of vast 
numbers of individual entities. But averages are pre- 
dictable because they are averages, irrespective of the 
type of government of the phenomena underlying 
them. 



PREDICTABILITY OF EVENTS 301 

Considering an atom alone in the world in State 3, 
the classical theory would have asked, and hoped to 
answer, the question, What will it do next? The quan- 
tum theory substitutes the question, Which will it do 
next? Because it admits only two lower states for the 
atom to go to. Further, it makes no attempt to find a 
definite answer, but contents itself with calculating the 
respective odds on the jumps to State 1 and State 2. 
The quantum physicist does not fill the atom with 
gadgets for directing its future behaviour, as the classical 
physicist would have done; he fills it with gadgets de- 
termining the odds on its future behaviour. He studies 
the art of the bookmaker not of the trainer. 

Thus in the structure of the world as formulated in 
the new quantum theory it is predetermined that of 
500 atoms now in State 3, approximately 400 will go 
on to State 1 and 100 to State 2 — in so far as anything 
subject to chance fluctuations can be said to be pre- 
determined. The odds of 4 to 1 find their appropriate 
representation in the picture of the atom; that is to say, 
something symbolic of a 4 : 1 ratio is present in each of 
the 500 atoms. But there are no marks distinguishing 
the atoms belonging to the group of 100 from the 400. 
Probably most physicists would take the view that 
although the marks are not yet shown in the picture, 
they are nevertheless present in Nature; they belong to 
an elaboration of the theory which will come in good 
time. The marks, of course, need not be in the atom 
itself; they may be in the environment which will 
interact with it. For example, we may load dice in such 
a way that the odds are 4 to 1 on throwing a 6. Both 
those dice which turn up 6 and those which do not 
have these odds written in their constitution — by a 
displaced position of the centre of gravity. The result 



302 CAUSATION 

of a particular throw is not marked in the dice; never- 
theless it is strictly causal (apart perhaps from the 
human element involved in throwing the dice) being de- 
termined by the external influences which are concerned. 
Our own position at this stage is that future develop- 
ments of physics may reveal such causal marks (either 
in the atom or in the influences outside it) or it may not. 
Hitherto whenever we have thought we have detected 
causal marks in natural phenomena they have always 
proved spurious, the apparent determinism having come 
about in another way. Therefore we are inclined to 
regard favourably the possibility that there may be no 
causal marks anywhere. 

But, it will be said, it is inconceivable that an atom 
can be so evenly balanced between two alternative 
courses that nowhere in the world as yet is there any 
trace of the ultimately deciding factor. This is an ap- 
peal to intuition and it may fairly be countered with 
another appeal to intuition. I have an intuition much 
more immediate than any relating to the objects of the 
physical world; this tells me that nowhere in the world 
as yet is there any trace of a deciding factor as to 
whether I am going to lift my right hand or my left. 
It depends on an unfettered act of volition not yet made 
or foreshadowed.* My intuition is that the future is 
able to bring forth deciding factors which are not 
secretly hidden in the past. 

The position is that the laws governing the micro- 
scopic elements of the physical world — individual 
atoms, electrons, quanta — do not make definite pre- 
dictions as to what the individual will do next. I am 

* It is fair to assume the trustworthiness of this intuition in answering 
an argument which appeals to intuition; the assumption would beg the 
question if we were urging the argument independently. 



THE NEW EPISTEMOLOGICAL OUTLOOK 303 

here speaking of the laws that have been actually dis- 
covered and formulated on the old quantum theory and 
the new. These laws indicate several possibilities in the 
future and state the odds on each. In general the odds 
are moderately balanced and are not tempting to an 
aspiring prophet. But short odds on the behaviour of 
individuals combine into very long odds on suitably 
selected statistics of a number of individuals; and the 
wary prophet can find predictions of this kind on which 
to stake his credit — without serious risk. All the success- 
ful predictions hitherto attributed to causality are trace- 
able to this. It is quite true that the quantum laws for 
individuals are not incompatible with causality; they 
merely ignore it. But if we take advantage of this 
indifference to reintroduce determinism at the basis of 
world structure it is because our philosophy predisposes 
us that way, not because we know of any experimental 
evidence in its favour. 

We might for illustration make a comparison with 
the doctrine of predestination. That theological doc- 
trine, whatever may be said against it, has hitherto 
seemed to blend harmoniously with the predetermination 
of the material universe. But if we were to appeal to 
the new conception of physical law to settle this question 
by analogy the answer would be : — The individual is not 
predestined to arrive at either of the two states, which 
perhaps may here be sufficiently discriminated as 
State 1 and State 2; the most that can be considered 
already settled is the respective odds on his reaching 
these states. 

The New Epistemological Outlook. Scientific investiga- 
tion does not lead to knowledge of the intrinsic nature 
of things. "Whenever we state the properties of a body 



304 CAUSATION 

in terms of physical quantities we are imparting know- 
ledge of the response of various metrical indicators to 
its presence and nothing more" (p. 257). But if a body- 
is not acting according to strict causality, if there is an 
element of uncertainty as to the response of the indica- 
tors, we seem to have cut away the ground for this kind of 
knowledge. It is not predetermined what will be the 
reading of the weighing-machine if the body is placed 
on it, therefore the body has no definite mass; nor where 
it will be found an instant hence, therefore it has no 
definite velocity; nor where the rays now being reflected 
from it will converge in the microscope, therefore it has 
no definite position; and so on. It is no use answering 
that the body really has a definite mass, velocity, 
position, etc., which we are unaware of; that statement, 
if it means anything, refers to an intrinsic nature of 
things outside the scope of scientific knowledge. We 
cannot infer these properties with precision from any- 
thing that we can be aware of, because the breach of 
causality has broken the chain of inference. Thus our 
knowledge of the response of indicators to the presence 
of the body is non-existent; therefore we cannot assert 
knowledge of it at all. So what is the use of talking 
about it? The body which was to be the abstraction of 
all these (as yet unsettled) pointer readings has become 
superfluous in the physical world. That is the dilemma 
into which the old epistemology leads us as soon as we 
begin to doubt strict causality. 

In phenomena on a gross scale this difficulty can be 
got round. A body may have no definite position but 
yet have within close limits an extremely probable 
position. When the probabilities are large the substitu- 
tion of probability for certainty makes little difference; 
it adds only a negligible haziness to the world. But 



THE NEW EPISTEMOLOGICAL OUTLOOK 305 

though the practical change is unimportant there are 
fundamental theoretical consequences. All probabilities 
rest on a basis of a priori probability, and we cannot say 
whether probabilities are large or small without having 
assumed such a basis. In agreeing to accept those of our 
calculated probabilities which are very high as virtually 
equivalent to certainties on the old scheme, we are as it 
were making our adopted basis of a priori probability 
a constituent of the world-structure — adding to the 
world a kind of symbolic texture that cannot be ex- 
pressed on the old scheme. 

On the atomic scale of phenomena the probabilities 
are in general well-balanced, and there are no "naps" 
for the scientific punter to put his shirt on. If a body is 
still defined as a bundle of pointer readings (or highly 
probable pointer readings) there are no "bodies" on 
the atomic scale. All that we can extract is a bundle of 
probabilities. That is in fact just how Schrodinger tries 
to picture the atom — as a wave centre of his probability 
entity i|>. 

We commonly have had to deal with probabilities 
which arise through ignorance. With fuller knowledge 
we should sweep away the references to probability and 
substitute the exact facts. But it appears to be a funda- 
mental point in Schrodinger's theory that his probabili- 
ties are not to be replaced in that way. When his ip is 
sufficiently concentrated it indicates the point where the 
electron is; when it is diffused it gives only a vague 
indication of the position. But this vague indication is 
not something which ideally ought to be replaced by 
exact knowledge; it is ip itself which acts as the source 
of the light emitted from the atom, the period of the 
light being that of the beats of i|>. I think this means 
that the spread of ty is not a symbol for uncertainty aris- 



306 CAUSATION 

ing through lack of information; it is a symbol for 
causal failure — an indeterminacy of behaviour which is 
part of the character of the atom. 

We have two chief ways of learning about the interior 
of the atom. We can observe electrons entering or 
leaving, and we can observe light entering or leaving. 
Bohr has assumed a structure connected by strictly 
causal law with the first phenomenon, Heisenberg and 
his followers with the second. If the two structures were 
identifiable then the atom w r ould involve a complete 
causal connection of the two types of phenomena. But 
apparently no such causal linkage exists. Therefore we 
have to be content with a correlation in which the 
entities of the one model represent probabilities in the 
second model. There are perhaps details in the two 
theories which do not quite square with this; but it 
seems to express the ideal to be aimed at in describing 
the laws of an incompletely causal world, viz. that the 
causal source of one phenomenon shall represent the 
probability of causal source of another phenomenon. 
Schrodinger's theory has given at least a strong hint 
that the actual world is controlled on this plan. 

The Principle of Indeterminacy. Thus far we have 
shown that modern physics is drifting away from the 
postulate that the future is predetermined, ignoring it 
rather than deliberately rejecting it. With the discovery 
of the Principle of Indeterminacy (p. 220) its attitude 
has become more definitely hostile. 

Let us take the simplest case in which we think we 
can predict the future. Suppose that we have a particle 
with known position and velocity at the present instant. 
Assuming that nothing interferes with it we can predict 
the position at a subsequent instant. (Strictly the non- 



THE PRINCIPLE OF INDETERMINACY 307 

interference would be a subject for another prediction, 
but to simplify matters we shall concede it.) It is just 
this simple prediction which the principle of indeter- 
minacy expressly forbids. It states that we cannot know 
accurately both the velocity and position of a particle 
at the present instant. 

At first sight there seems to be an inconsistency. 
There is no limit to the accuracy with which we may 
know the position, provided that we do not want to 
know the velocity also. Very well; let us make a highly 
accurate determination of position now, and after 
waiting a moment make another highly accurate deter- 
mination of position. Comparing the two accurate 
positions we compute the accurate velocity — and snap 
our fingers at the principle of indeterminacy. This 
velocity, however, is of no use for prediction, because in 
making the second accurate determination of position 
we have rough-handled the particle so much that it no 
longer has the velocity we calculated. // is a purely 
retrospective velocity. The velocity does not exist in the 
present tense but in the future perfect; it never exists, 
it never will exist, but a time may come when it will have 
existed. There is no room for it in Fig. 4 which contains 
an Absolute Future and an Absolute Past but not an 
Absolute Future Perfect. 

The velocity which we attribute to a particle now 
can be regarded as an anticipation of its future positions. 
To say that it is unknowable (except with a certain 
degree of inaccuracy) is to say that the future cannot be 
anticipated. Immediately the future is accomplished, 
so that it is no longer an anticipation, the velocity be- 
comes knowable. 

The classical view that a particle necessarily has a 
definite (but not necessarily knowable) velocity now, 



308 CAUSATION 

amounts to disguising a piece of the unknown future as 
an unknowable element of the present. Classical physics 
foists a deterministic scheme on us by a trick; it smuggles 
the unknown future into the present, trusting that we 
shall not press an inquiry as to whether it has become 
any more knowable that way. 

The same principle extends to every kind of pheno- 
menon that we attempt to predict, so long as the need 
for accuracy is not buried under a mass of averages. To 
every co-ordinate there corresponds a momentum, and 
by the principle of indeterminacy the more accurately 
the co-ordinate is known the less accurately the momen- 
tum is known. Nature thus provides that knowledge 
of one-half of the world will ensure ignorance of the 
other half — ignorance which, we have seen, may be 
remedied later when the same part of the world is con- 
templated retrospectively. We can scarcely rest content 
with a picture of the world which includes so much that 
cannot be known. We have been trying to get rid of 
unknowable things, i.e. all conceptions which have no 
causal connection with our experience. We have elimi- 
nated velocity through aether, "right" frames of space, 
etc., for this reason. This vast new unknowable element 
must likewise be swept out of the Present. Its proper 
place is in the Future because then it will no 
longer be unknowable. It has been put in prematurely 
as an anticipation of that which cannot be antici- 
pated. 

In assessing whether the symbols which the physicist 
has scattered through the external world are adequate to 
predetermine the future, we must be on our guard 
against retrospective symbols. It is easy to prophesy 
after the event. 



NATURAL AND SUPERNATURAL 309 

Natural and Supernatural. A rather serious consequence 
of dropping causality in the external world is that it 
leaves us with no clear distinction between the Natural 
and the Supernatural. In an earlier chapter I compared 
the invisible agent invented to account for the tug of 
gravitation to a "demon". Is a view of the world which 
admits such an agent any more scientific than that of a 
savage who attributes all that he finds mysterious in 
Nature to the work of invisible demons? The New- 
tonian physicist had a valid defence. He could point 
out that his demon Gravitation was supposed to act 
according to fixed causal laws and was therefore not to 
be compared with the irresponsible demons of the 
savage. Once a deviation from strict causality is ad- 
mitted the distinction melts away. I suppose that the 
savage would admit that his demon was to some extent 
a creature of habit and that it would be possible to make 
a fair guess as to what he would do in the future; but 
that sometimes he would show a will of his own. It is 
that imperfect consistency which formerly disqualified 
him from admission as an entity of physics along with 
his brother Gravitation. 

That is largely why there has been so much bother 
about "me"; because I have, or am persuaded that I 
have, "a will of my own". Either the physicist must 
leave his causal scheme at the mercy of supernatural 
interference from me, or he must explain away my 
supernatural qualities. In self-defence the materialist 
favoured the latter course; he decided that I was not 
supernatural — only complicated. We on the other hand 
have concluded that there is no strict causal behaviour 
anywhere. We can scarcely deny the charge that in 
abolishing the criterion of causality we are opening the 
door to the savage's demons. It is a serious step, but 



310 CAUSATION 

I do not think it means the end of all true science. After 
all if they try to enter we can pitch them out again, as 
Einstein pitched out the respectable causal demon who 
called himself Gravitation. It is a privation to be no 
longer able to stigmatise certain views as unscientific 
superstition; but we are still allowed, if the circumstances 
justify it, to reject them as bad science. 

Volition. From the philosophic point of view it is of deep 
interest to consider how this affects the freedom of the 
human mind and spirit. A complete determinism of 
the material universe cannot be divorced from deter- 
minism of the mind. Take, for example, the prediction 
of the weather this time next year. The prediction is 
not likely ever to become practicable, but "orthodox" 
physicists are not yet convinced that it is theoretically 
impossible; they hold that next year's weather is already 
predetermined. We should require extremely detailed 
knowledge of present conditions, since a small local 
deviation can exert an ever-expanding influence. 
We must examine the state of the sun so as to predict 
the fluctuations in the heat and corpuscular radiation 
which it sends us. We must dive into the bowels of the 
earth to be forewarned of volcanic eruptions which may 
spread a dust screen over the atmosphere as Mt. Katmai 
did some years ago. But further we must penetrate into 
the recesses of the human mind. A coal strike, a great 
war, may directly change the conditions of the atmo- 
sphere; a lighted match idly thrown away may cause 
deforestation which will change the rainfall and climate. 
There can be no fully deterministic control of inorganic 
phenomena unless the determinism governs mind itself. 
Conversely if we wish to emancipate mind we must to 
some extent emancipate the material world also. There 



VOLITION 311 

appears to be no longer any obstacle to this emanci- 
pation. 

Let us look more closely into the problem of how the 
mind gets a grip on material atoms so that movements 
of the body and limbs can be controlled by its volition. 
I think we may now feel quite satisfied that the volition 
is genuine. The materialist view was that the motions 
which appear to be caused by our volition are really 
reflex actions controlled by the material processes in the 
brain, the act of will being an inessential side pheno- 
menon occurring simultaneously with the physical 
phenomena. But this assumes that the result of apply- 
ing physical laws to the brain is fully determinate. It is 
meaningless to say that the behaviour of a conscious 
brain is precisely the same as that of a mechanical brain 
if the behaviour of a mechanical brain is left undeter- 
mined. If the laws of physics are not strictly causal the 
most that can be said is that the behaviour of the 
conscious brain is one of the possible behaviours of a 
mechanical brain. Precisely so; and the decision between 
the possible behaviours is what we call volition. 

Perhaps you will say, When the decision of an atom 
is made between its possible quantum jumps, is that 
also "volition"? Scarcely; the analogy is altogether too 
remote. The position is that both for the brain and the 
atom there, is nothing in the physical world, i.e. the 
world of pointer readings, to predetermine the decision; 
the decision is a fact of the physical world with con- 
sequences in the future but not causally connected to 
the past. In the case of the brain we have an insight 
into a mental world behind the world of pointer readings 
and in that world we get a new picture of the fact of 
decision which must be taken as revealing its real 
nature — if the words real nature have any meaning. 



312 CAUSATION 

For the atom we have no such insight into what is 
behind the pointer readings. We believe that behind 
all pointer readings there is a background continuous 
with the background of the brain; but there is no more 
ground for calling the background of the spontaneous 
behaviour of the atom "volition" than for calling the 
background of its causal behaviour "reason". It should 
be understood that we are not attempting to reintroduce 
in the background the strict causality banished from 
the pointer readings. In the one case in which we have 
any insight — the background of the brain — we have 
no intention of giving up the freedom of the mind and 
will. Similarly we do not suggest that the marks of 
predestination of the atom, not found in the pointer 
readings, exist undetectable in the unknown back- 
ground. To the question whether I would admit that 
the cause of the decision of the atom has something in 
common with the cause of the decision of the brain, 
I would simply answer that there is no cause. In the 
case of the brain I have a deeper insight into the 
decision; this insight exhibits it as volition, i.e. some- 
thing outside causality. 

A mental decision to turn right or turn left starts one 
of two alternative sets of impulses along the nerves to 
the feet. At some brain centre the course of behaviour 
of certain atoms or elements of the physical world is 
directly determined for them by the mental decision — 
or, one may say, the scientific description of that be- 
haviour is the metrical aspect of the decision. It would 
be a possible though difficult hypothesis to assume that 
very few atoms (or possibly only one atom) have this 
direct contact with the conscious decision, and that 
these few atoms serve as a switch to deflect the material 
world from one course to the other. But it is physically 



MIND AND STATISTICAL LAWS 313 

improbable that each atom has its duty in the brain so 
precisely allotted that the control of its behaviour would 
prevail over all possible irregularities of the other atoms. 
If I have at all rightly understood the processes of my 
own mind, there is no finicking with individual atoms. 
I do not think that our decisions are precisely 
balanced on the conduct of certain key-atoms. Could 
we pick out one atom in Einstein's brain and say that 
if it had made the wrong quantum jump there would 
have been a corresponding flaw in the theory of rela- 
tivity? Having regard to the physical influences of 
temperature and promiscuous collision it is impossible 
to maintain this. It seems that we must attribute to the 
mind power not only to decide the behaviour of atoms 
individually but to affect systematically large groups — 
in fact to tamper with the odds on atomic behaviour. 
This has always been one of the most dubious points 
in the theory of the interaction of mind and matter. 

Interference with Statistical Laws. Has the mind power 
to set aside statistical laws which hold in inorganic 
matter? Unless this is granted its opportunity of inter- 
ference seems to be too circumscribed to bring about 
the results which are observed to follow from mental 
decisions. But the admission involves a genuine 
physical difference between inorganic and organic (or, 
at any rate, conscious) matter. I would prefer to avoid 
this hypothesis, but it is necessary to face the issue 
squarely. The indeterminacy recognised in modern 
quantum theory is only a partial step towards freeing 
our actions from deterministic control. To use an 
analogy — we have admitted an uncertainty which may 
take or spare human lives; but we have yet to find an 
uncertainty which may upset the expectations of a life- 



3H CAUSATION 

insurance company. Theoretically the one uncertainty 
might lead to the other, as when the fate of millions 
turned on the murders at Sarajevo. But the hypothesis 
that the mind operates through two or three key-atoms 
in the brain is too desperate a way of escape for us, and 
I reject it for the reasons already stated. 

It is one thing to allow the mind to direct an atom 
between two courses neither of which would be im- 
probable for an inorganic atom; it is another thing to 
allow it to direct a crowd of atoms into a configuration 
which the secondary laws of physics would set aside as 
"too improbable". Here the improbability is that a 
large number of entities each acting independently 
should conspire to produce the result; it is like the 
improbability of the atoms finding themselves by chance 
all in one half of a vessel. We must suppose that in the 
physical part of the brain immediately affected by a 
mental decision there is some kind of interdependence 
of behaviour of the atoms which is not present in 
inorganic matter. 

I do not wish to minimise the seriousness of admitting 
this difference between living and dead matter. But 
I think that the difficulty has been eased a little, if it 
has not been removed. To leave the atom constituted as 
it was but to interfere with the probability of its un- 
determined behaviour, does not seem quite so drastic 
an interference with natural law as other modes of 
mental interference that have been suggested. (Perhaps 
that is only because we do not understand enough about 
these probabilities to realise the heinousness of our 
suggestion.) Unless it belies its name, probability can 
be modified in ways which ordinary physical entities 
would not admit of. There can be no unique probability 
attached to any event or behaviour; we can only speak 



MIND AND STATISTICAL LAWS 315 

of "probability in the light of certain given informa- 
tion", and the probability alters according to the extent 
of the information. It is, I think, one of the most un- 
satisfactory features of the new quantum theory in its 
present stage that it scarcely seems to recognise this 
fact, and leaves us to guess at the basis of information 
to which its probability theorems are supposed to refer. 

Looking at it from another aspect — if the unity of 
a man's consciousness is not an illusion, there must be 
some corresponding unity in the relations of the mind- 
stuff which is behind the pointer readings. Applying 
our measures of relation structure, as in chapter XI, 
we shall build matter and fields of force obeying 
identically the principal field-laws; the atoms will 
individually be in no way different from those which 
are without this unity in the background. But it seems 
plausible that when we consider their collective be- 
haviour we shall have to take account of the broader 
unifying trends in the mind-stuff, and not expect the 
statistical results to agree with those appropriate to 
structures of haphazard origin. 

I think that even a materialist must reach a conclusion 
not unlike ours if he fairly faces the problem. He will 
need in the physical world something to stand for a 
symbolic unity of the atoms associated with an individual 
consciousness, which does not exist for atoms not so 
associated — a unity which naturally upsets physical 
predictions abased on the hypothesis of random dis- 
connection. For he has not only to translate into 
material configurations the multifarious thoughts and 
images of the mind, but must surely not neglect to find 
some kind of physical substitute for the Ego. 



Chapter XV 

SCIENCE AND MYSTICISM 

One day I happened to be occupied with the subject 

of "Generation of Waves by Wind". I took down the 

standard treatise on hydrodynamics, and under that 
heading I read — 

The equations (12) and (13) of the preceding Art. enable us 
to examine a related question of some interest, viz. the generation 
and maintenance of waves against viscosity, by suitable forces 
applied to the surface. 

If the external forces p' yy , p'^ be given multiples of «***+**, 
where k and a are prescribed, the equations in question determine 
A and C, and thence, by (9) the value of tj. Thus we find 

P'vv _ (^ + 2yffflS + Q 2 ) A - i ((T 2 + 2vkma) C 
gprj "" gk(J- iC) l 

£*v___a 2hk 2 J + (a + 2yg| C 
gprj-gk' (J-iQ " 

where o 2 has been written for gk -\- T r k z as before. . . . 

And so on for two pages. At the end it is made clear 
that a wind of less than half a mile an hour will leave 
the surface unruffled. At a mile an hour the surface is 
covered with minute corrugations due to capillary waves 
which decay immediately the disturbing cause ceases. 
At two miles an hour the gravity waves appear. As 
the author modestly concludes, "Our theoretical investi- 
gations give considerable insight into the incipient stages 
of wave-formation". 

On another occasion the same subject of "Generation 

316 



SCIENCE AND MYSTICISM 317 

of Waves by Wind" was in my mind; but this time 
another book was more appropriate, and I read — 

There are waters blown by changing winds to laughter 
And lit by the rich skies, all day. And after, 

Frost, with a gesture, stays the waves that dance 
And wandering loveliness. He leaves a white 

Unbroken glory, a gathered radiance, 
A width, a shining peace, under the night. 

The magic words bring back the scene. Again we 
feel Nature drawing close to us, uniting with us, till 
we are filled with the gladness of the waves dancing in 
the sunshine, with the awe of the moonlight on the 
frozen lake. These were not moments when we fell 
below ourselves. We do not look back on them and say, 
"It was disgraceful for a man with six sober senses and 
a scientific understanding to let himself be deluded in 
that way. I will take Lamb's Hydrodynamics with me 
next time". It is good that there should be such 
moments for us. Life would be stunted and narrow if 
we could feel no significance in the world around us 
beyond that which can be weighed and measured with 
the tools of the physicist or described by the metrical 
symbols of the mathematician. 

Of course it was an illusion. We can easily expose 
the rather clumsy trick that was played on us. Aethereal 
vibrations of various wave-lengths, reflected at different 
angles from the disturbed interface between air and 
water, reached our eyes, and by photoelectric action 
caused appropriate stimuli to travel along the optic 
nerves to a brain-centre. Here the mind set to work to 
weave an impression out of the stimuli. The incoming 
material was somewhat meagre; but the mind is a great 
storehouse of associations that could be used to clothe 



318 SCIENCE AND MYSTICISM 

the skeleton. Having woven an impression the mind 
surveyed all that it had made and decided that it was 
very good. The critical faculty was lulled. We ceased 
to analyse and were conscious only of the impression 
as a whole. The warmth of the air, the scent of the 
grass, the gentle stir of the breeze, combined with the 
visual scene in one transcendent impression, around us 
and within us. Associations emerging from their store- 
house grew bolder. Perhaps we recalled the phrase 
"rippling laughter". Waves — ripples — laughter — glad- 
ness — the ideas jostled one another. Quite illogically we 
were glad; though what there can possibly be to be glad 
about in a set of aethereal vibrations no sensible person 
can explain. A mood of quiet joy suffused the whole 
impression. The gladness in ourselves was in Nature, 
in the waves, everywhere. That's how it was. 

It was an illusion. Then why toy with it longer? 
These airy fancies which the mind, when we do not 
keep it severely in order, projects into the external world 
should be of no concern to the earnest seeker after truth. 
Get back to the solid substance of things, to the material 
of the water moving under the pressure of the wind and 
the force of gravitation in obedience to the laws of 
hydrodynamics. But the solid substance of things is 
another illusion. It too is a fancy projected by the mind 
into the external world. We have chased the solid 
substance from the continuous liquid to the atom, from 
the atom to the electron, and there we have lost it. But 
at least, it will be said, we have reached something real 
at the end of the chase — the protons and electrons. Or 
if the new quantum theory condemns these images as 
too concrete and leaves us with no coherent images at 
all, at least we have symbolic co-ordinates and momenta 
and Hamiltonian functions devoting themselves with 



SCIENCE AND MYSTICISM 319 

single-minded purpose to ensuring that qp — pq shall be 
equal to ih/m. 

In a previous chapter I have tried to show that by 
following this course we reach a cyclic scheme which 
from its very nature can only be a partial expression of 
our environment. It is not reality but the skeleton of 
reality. "Actuality" has been lost in the exigencies of 
the chase. Having first rejected the mind as a worker 
of illusion we have in the end to return to the mind and 
say, "Here are worlds well and truly built on a basis 
more secure than your fanciful illusions. But there is 
nothing to make any one of them an actual world. 
Please choose one and weave your fanciful images into 
it. That alone can make it actual". We have torn away 
the mental fancies to get at the reality beneath, only to 
find that the reality of that which is beneath is bound 
up with its potentiality of awakening these fancies. It 
is because the mind, the weaver of illusion, is also the 
only guarantor of reality that reality is always to be 
sought at the base of illusion. Illusion is to reality as 
the smoke to the fire. I will not urge that hoary un- 
truth "There is no smoke without fire". But it is 
reasonable to inquire whether in the mystical illusions of 
man there is not a reflection of an underlying reality. 

To put a plain question — Why should it be good for 
us to experience a state of self-deception such as I have 
described? I think everyone admits that it is good to 
have a spirit sensitive to the influences of Nature, good 
to exercise an appreciative imagination and not always 
to be remorselessly dissecting our environment after 
the manner of the mathematical physicists. And it is 
good not merely in a utilitarian sense, but in some 
purposive sense necessary to the fulfilment of the life 
that is given us. It is not a dope which it is expedient 



320 SCIENCE AND MYSTICISM 

to take from time to time so that we may return with 
greater vigour to the more legitimate employment of 
the mind in scientific investigation. Just possibly it 
might be defended on the ground that it affords to the 
non-mathematical mind in some feeble measure that 
delight in the external world which would be more 
fully provided by an intimacy with its differential 
equations. (Lest it should be thought that I have 
intended to pillory hydrodynamics, I hasten to say in 
this connection that I would not rank the intellectual 
(scientific) appreciation on a lower plane than the 
mystical appreciation; and I know of passages written 
in mathematical symbols which in their sublimity might 
vie with Rupert Brooke's sonnet.) But I think you will 
agree with me that it is impossible to allow that the one 
kind of appreciation can adequately fill the place of the 
other. Then how can it be deemed good if there is 
nothing in it but self-deception? That would be an 
upheaval of all our ideas of ethics. It seems to me that 
the only alternatives are either to count all such sur- 
render to the mystical contact of Nature as mischievous 
and ethically wrong, or to admit that in these moods 
we catch something of the true relation of the world to 
ourselves — a relation not hinted at in a purely scientific 
analysis of its content. I think the most ardent material- 
ist does not advocate, or at any rate does not practice, 
the first alternative; therefore I assume the second alter- 
native, that there is some kind of truth at the base of the 
illusion. 

But we must pause to consider the extent of the 
illusion. Is it a question of a small nugget of reality 
buried under a mountain of illusion? If that were so it 
would be our duty to rid our minds of some of the 
illusion at least, and try to know the truth in purer form. 



SYMBOLIC AND INTIMATE KNOWLEDGE 321 

But I cannot think there is much amiss with our appre- 
ciation of the natural scene that so impresses us. I do 
not think a being more highly endowed than ourselves 
would prune away much of what we feel. It is not so 
much that the feeling itself is at fault as that our 
introspective examination of it wraps it in fanciful 
imagery. If I were to try to put into words the essen- 
tial truth revealed in the mystic experience, it would be 
that our minds are not apart from the world; and the 
feelings that we have of gladness and melancholy and 
our yet deeper feelings are not of ourselves alone, but 
are glimpses of a reality transcending the narrow limits 
of our particular consciousness — that the harmony and 
beauty of the face of Nature is at root one with the 
gladness that transfigures the face of man. We try to 
express much the same truth when we say that the 
physical entities are only an extract of pointer readings 
and beneath them is a nature continuous with our own. 
But I do not willingly put it into words or subject it to 
introspection. We have seen how in the physical world 
the meaning is greatly changed when we contemplate 
it as surveyed from without instead of, as it essentially 
must be, from within. By introspection we drag out the 
truth for external survey; but in the mystical feeling 
the truth is apprehended from within and is, as it should 
be, a part of ourselves. 

Symbolic Knowledge and Intimate Knowledge. May I 
elaborate this objection to introspection? We have two 
kinds of knowledge which I call symbolic knowledge 
and intimate knowledge. I do not know whether it 
would be correct to say that reasoning is only applicable 
to symbolic knowledge, but the more customary forms 
of reasoning have been developed for symbolic know- 



322 SCIENCE AND MYSTICISM 

ledge only. The intimate knowledge will not submit to 
codification and analysis; or, rather, when we attempt 
to analyse it the intimacy is lost and it is replaced by 
symbolism. 

For an illustration let us consider Humour. I suppose 
that humour can be analysed to some extent and the 
essential ingredients of the different kinds of wit 
classified. Suppose that we are offered an alleged joke. 
We subject it to scientific analysis as we would a chemical 
salt of doubtful nature, and perhaps after careful con- 
sideration of all its aspects we are able to confirm that 
it really and truly is a joke. Logically, I suppose, our 
next procedure would be to laugh. But it may certainly 
be predicted that as the result of this scrutiny we shall 
have lost all inclination we may ever have had to laugh 
at it. It simply does not do to expose the inner workings 
of a joke. The classification concerns a symbolic know- 
ledge of humour which preserves all the characteristics 
of a joke except its laughableness. The real appreciation 
must come spontaneously, not introspectively. I think 
this is a not unfair analogy for our mystical feeling for 
Nature, and I would venture even to apply it to our 
mystical experience of God. There are some to whom 
the sense of a divine presence irradiating the soul is one 
of the most obvious things of experience. In their view 
a man without this sense is to be regarded as we regard 
a man without a sense of humour. The absence is a kind 
of mental deficiency. We may try to analyse the ex- 
perience as we analyse humour, and construct a theology, 
or it may be an atheistic philosophy, which shall put 
into scientific form what is to be inferred about it. But 
let us not forget that the theology is symbolic knowledge 
whereas the experience is intimate knowledge. And as 
laughter cannot be compelled by the scientific exposition 



DEFENCE OF MYSTICISM 323 

of the structure of a joke, so a philosophic discussion 
of the attributes of God (or an impersonal substitute) 
is likely to miss the intimate response of the spirit which 
is the central point of the religious experience. 

Defence of Mysticism. A defence of the mystic might 
run something like this. We have acknowledged that the 
entities of physics can from their very nature form only 
a partial aspect of the reality. How are we to deal with 
the other part? It cannot be said that that other part 
concerns us less than the physical entities. Feelings, 
purpose, values, make up our consciousness as much as 
sense-impressions. We follow up the sense-impressions 
and find that they lead into an external world discussed 
by science; we follow up the other elements of our 
being and find that they lead — not into a world of space 
and time, but surely somewhere. If you take the view 
that the whole of consciousness is reflected in the dance 
of electrons in the brain, so that each emotion is a 
separate figure of the dance, then all features of con- 
sciousness alike lead into the external world of physics. 
But I assume that you have followed me in rejecting 
this view, and that you agree that consciousness as a 
whole is greater than those quasi-metrical aspects of it 
which are abstracted to compose the physical brain. 
We have then to deal with those parts of our being 
unamenable to metrical specification, that do not make 
contact — jut out, as it were — into space and time. By 
dealing with them I do not mean make scientific in- 
quiry into them. The first step is to give acknowledged 
status to the crude conceptions in which the mind invests 
them, similar to the status of those crude conceptions 
which constitute the familiar material world. 

Our conception of the familiar table was an illusion. 



324 SCIENCE AND MYSTICISM 

But if some prophetic voice had warned us that it was 
an illusion and therefore we had not troubled to investi- 
gate further we should never have found the scientific 
table. To reach the reality of the table we need to be 
endowed with sense-organs to weave images and illusions 
about it. And so it seems to me that the first step in a 
broader revelation to man must be the awakening of 
image-building in connection with the higher faculties 
of his nature, so that these are no longer blind alleys 
but open out into a spiritual world — a world partly of 
illusion, no doubt, but in which he lives no less than in 
the world, also of illusion, revealed by the senses. 

The mystic, if haled before a tribunal of scientists, 
might perhaps end his defence on this note. He would 
say, The familiar material world of everyday conception, 
though lacking somewhat in scientific truth, is good 
enough to live in; in fact the scientific world of pointer 
readings would be an impossible sort of place to inhabit. 
It is a symbolic world and the only thing that could live 
comfortably in it would be a symbol. But I am not 
a symbol; I am compounded of that mental activity 
which is from your point of view a nest of illusion, so 
that to accord with my own nature I have to transform 
even the world explored by my senses. But I am not 
merely made up of senses; the rest of my nature has to 
live and grow. I have to render account of that environ- 
ment into which it has its outlet. My conception of 
my spiritual environment is not to be compared with 
your scientific world of pointer readings; it is an every- 
day world to be compared with the material world of 
familiar experience. I claim it as no more real and no 
less real than that. Primarily it is not a world to be 
analysed, but a world to be lived in." 

Granted that this takes us outside the sphere of 



DEFENCE OF MYSTICISM 325 

exact knowledge, and that it is difficult to imagine that 
anything corresponding to exact science will ever be 
applicable to this part of our environment, the mystic 
is unrepentant. Because we are unable to render exact 
account of our environment it does not follow that it 
would be better to pretend that we live in a vacuum. 

If the defence may be considered to have held good 
against the first onslaught, perhaps the next stage of the 
attack will be an easy tolerance. "Very well. Have it 
your own way. It is a harmless sort of belief — not like 
a more dogmatic theology. You want a sort of spiritual 
playground for those queer tendencies in man's nature, 
which sometimes take possession of him. Run away 
and play then; but do not bother the serious people who 
are making the world go round." The challenge now 
comes not from the scientific materialism which pro- 
fesses to seek a natural explanation of spiritual power, 
but from the deadlier moral materialism which despises 
it. Few deliberately hold the philosophy that the forces 
of progress are related only to the material side of our 
environment, but few can claim that they are not more 
or less under its sway. We must not interrupt the 
"practical men", these busy moulders of history carry- 
ing us at ever-increasing pace towards our destiny as 
an ant-heap of humanity infesting the earth. But is it 
true in history that material forces have been the 
most potent factors? Call it of God, of the Devil, 
fanaticism, unreason; but do not underrate the power of 
the mystic. Mysticism may be fought as error or believed 
as inspired, but it is no matter for easy tolerance — 

We are the music-makers 

And we are the dreamers of dreams 

Wandering by lone sea-breakers 
And sitting by desolate streams ; 



326 SCIENCE AND MYSTICISM 

World-losers and world-forsakers, 

On whom the pale moon gleams: 
Yet we are the movers and shakers 

Of the world for ever, it seems. 

Reality and Mysticism, But a defence before the scien- 
tists may not be a defence to our own self-questionings. 
We are haunted by the word reality. I have already tried 
to deal with the questions which arise as to the meaning 
of reality; but it presses on us so persistently that, at the 
risk of repetition, I must consider it once more from 
the standpoint of religion. A compromise of illusion 
and reality may be all very well in our attitude towards 
physical surroundings; but to admit such a compromise 
into religion would seem to be a trifling with sacred 
things. Reality seems to concern religious beliefs much 
more than any others. No one bothers as to whether 
there is a reality behind humour. The artist who tries 
to bring out the soul in his picture does not really care 
whether and in what sense the soul can be said to exist. 
Even the physicist is unconcerned as to whether atoms 
or electrons really exist; he usually asserts that they do, 
but, as we have seen, existence is there used in a 
domestic sense and no inquiry is made as to whether 
it is more than a conventional term. In most subjects 
(perhaps not excluding philosophy) it seems sufficient 
to agree on the things that we shall call real, and after- 
wards try to discover what we mean by the word. And 
so it comes about that religion seems to be the one field 
of inquiry in which the question of reality and existence 
is treated as of serious and vital importance. 

But it is difficult to see how such an inquiry can be 
profitable. When Dr. Johnson felt himself getting tied 
up in argument over "Bishop Berkeley's ingenious 
sophistry to prove the non-existence of matter, and that 



REALITY AND MYSTICISM 327 

everything in the universe is merely ideal", he answered, 
"striking his foot with mighty force against a large 
stone, till he rebounded from it, — 'I refute it thus* ". 
Just what that action assured him of is not very obvious; 
but apparently he found it comforting. And to-day the 
matter-of-fact scientist feels the same impulse to recoil 
from these flights of thought back to something kick- 
able, although he ought to be aware by this time that 
what Rutherford has left us of the large stone is scarcely 
worth kicking. 

There is still the tendency to use "reality" as a word 
of magic comfort like the blessed word "Mesopotamia". 
If I were to assert the reality of the soul or of God, 
I should certainly not intend a comparison with 
Johnson's large stone — a patent illusion — or even with 
the p's and qs of the quantum theory — an abstract 
symbolism. Therefore I have no right to use the word 
in religion for the purpose of borrowing on its behalf 
that comfortable feeling which (probably wrongly) has 
become associated with stones and quantum co-ordi- 
nates. 

Scientific instincts warn me that any attempt to 
answer the question "What is real?" in a broader sense 
than that adopted for domestic purposes in science, is 
likely to lead to a floundering among vain words and 
high-sounding epithets. We all know that there are 
regions of the human spirit untrammelled by the world 
of physics. In the mystic sense of the creation around 
us, in the expression of art, in a yearning towards God, 
the soul grows upward and finds the fulfilment of 
something implanted in its nature. The sanction for 
this development is within us, a striving born with our 
consciousness or an Inner Light proceeding from a 
greater power than ours. Science can scarcely question 



328 SCIENCE AND MYSTICISM 

this sanction, for the pursuit of science springs from a 
striving which the mind is impelled to follow, a ques- 
tioning that will not be suppressed. Whether in the 
intellectual pursuits of science or in the mystical pur- 
suits of the spirit, the light beckons ahead and the 
purpose surging in our nature responds. Can we not 
leave it at that? Is it really necessary to drag in the 
comfortable word "reality" to be administered like a 
pat on the back? 

The problem of the scientific world is part of a 
broader problem — the problem of all experience. Ex- 
perience may be regarded as a combination of self 
and environment, it being part of the problem to 
disentangle these two interacting components. Life, 
religion, knowledge, truth are all involved in this 
problem, some relating to the finding of ourselves, some 
to the finding of our environment from the experience 
confronting us. All of us in our lives have to make 
something of this problem; and it is an important 
condition that we who have to solve the problem are 
ourselves part of the problem. Looking at the very 
beginning, the initial fact is the feeling of purpose in 
ourselves which urges us to embark on the problem. 
We are meant to fulfil something by our lives. There 
are faculties with which we are endowed, or which we 
ought to attain, which must find a status and an outlet 
in the solution. It may seem arrogant that we should in 
this way insist on moulding truth to our own nature; 
but it is rather that the problem of truth can only spring 
from a desire for truth which is in our nature. 

A rainbow described in the symbolism of physics is 
a band of aethereal vibrations arranged in systematic 
order of wave-length from about -000040 cm. to 
•000072 cm. From one point of view we are paltering 



SIGNIFICANCE AND VALUES 329 

with the truth whenever we admire the gorgeous bow 
of colour, and should strive to reduce our minds to such 
a state that we receive the same impression from the 
rainbow as from a table of wave-lengths. But although 
that is how the rainbow impresses itself on an impersonal 
spectroscope, we are not giving the whole truth and 
significance of experience — the starting-point of the 
problem — if we suppress the factors wherein we our- 
selves differ from a spectroscope. We cannot say that 
the rainbow, as part of the world, was meant to convey 
the vivid effects of colour; but we can perhaps say that 
the human mind as part of the world was meant to 
perceive it that way. 

Significance and Values. When we think of the sparkling 
waves as moved with laughter we are evidently attri- 
buting a significance to the scene which was not there. 
The physical elements of the water — the scurrying 
electric charges — were guiltless of any intention to 
convey the impression that they were happy. But so 
also were they guiltless of any intention to convey the 
impression of substance, of colour, or of geometrical 
form of the waves. If they can be held to have had any 
intention at all it was to satisfy certain differential 
equations — and that was because they are the creatures 
of the mathematician who has a partiality for differential 
equations. The physical no less than the mystical 
significance of the scene is not there; it is here — in the 
mind. 

What we make of the world must be largely de- 
pendent on the sense-organs that we happen to possess. 
How the world must have changed since man came to 
rely on his eyes rather than his nose ! You are alone on 
the mountains wrapt in a great silence ; but equip yourself 



330 SCIENCE AND MYSTICISM 

with an extra artificial sense-organ and, lo! the aether is 
hideous with the blare of the Savoy bands. Or — 

The isle is full of noises, 
Sounds, and sweet airs, that give delight, and hurt not. 
Sometimes a thousand twangling instruments 
Will hum about mine ears ; and sometimes voices. 

So far as broader characteristics are concerned wc 
see in Nature what we look for or are equipped to look 
for. Of course, I do not mean that we can arrange the 
details of the scene; but by the light and shade of our 
values we can bring out things that shall have the broad 
characteristics we esteem. In this sense the value placed 
on permanence creates the world of apparent substance; 
in this sense, perhaps, the God within creates the God 
in Nature. But no complete view can be obtained so 
long as we separate our consciousness from the world 
of which it is a part. We can only speak speculatively 
of that which I have called the "background of the 
pointer readings"; but it would at least seem plausible 
that if the values which give the light and shade of the 
world are absolute they must belong to the background, 
unrecognised in physics because they are not in the 
pointer readings but recognised by consciousness which 
has its roots in the background. I have no wish to put 
that forward as a theory; it is only to emphasise that, 
limited as we are to a knowledge of the physical world 
and its points of contact with the background in isolated 
consciousness, we do not quite attain that thought of 
the unity of the whole which is essential to a complete 
theory. Presumably human nature has been specialised 
to a considerable extent by the operation of natural 
selection; and it might well be debated whether its 
valuation of permanence and other traits now apparently 



SIGNIFICANCE AND VALUES 331 

fundamental are essential properties of consciousness or 
have been evolved through interplay with the external 
world. In that case the values given by mind to the 
external world have originally come to it from the 
external world-stuff. Such a tossing to and fro of values 
is, I think, not foreign to our view that the world-stuff 
behind the pointer readings is of nature continuous with 
the mind. 

In viewing the world in a practical way values for 
normal human consciousness may be taken as standard. 
But the evident possibility of arbitrariness in this 
valuation sets us hankering after a standard that could 
be considered final and absolute. We have two alter- 
natives. Either there are no absolute values, so that the 
sanctions of the inward monitor in our consciousness are 
the final court of appeal beyonu which it is idle to in- 
quire. Or there are absolute values; then we can only 
trust optimistically that our values are some pale 
reflection of those of the Absolute Valuer, or that we 
have insight into the mind of the Absolute from whence 
come those strivings and sanctions whose authority we 
usually forbear to question. 

I have naturally tried to make the outlook reached in 
these lectures as coherent as possible, but I should not 
be greatly concerned if under the shafts of criticism it 
becomes very ragged. Coherency goes with finality; 
and the anxious question is whether our arguments have 
begun right rather than whether^ they have had the good 
fortune to end right. The leading points which have 
seemed to me to deserve philosophic consideration may 
be summarised as follows: 

(1) The symbolic nature of the entities of physics 
is generally recognised; and the scheme of physics is 
now formulated in such a way as to make it almost 



332 SCIENCE AND MYSTICISM 

self-evident that it is a partial aspect of something 
wider. 

(2) Strict causality is abandoned in the material 
world. Our ideas of the controlling laws are in process 
of reconstruction and it is not possible to predict what 
kind of form they will ultimately take; but all the in- 
dications are that strict causality has dropped out 
permanently. This relieves the former necessity of 
supposing that mind is subject to deterministic law or 
alternatively that it can suspend deterministic law in the 
material world. 

(3) Recognising that the physical world is entirely 
abstract and without "actuality" apart from its linkage 
to consciousness, we restore consciousness to the funda- 
mental position instead of representing it as an in- 
essential complication occasionally found in the midst 
of inorganic nature at a late stage of evolutionary 
history. 

(4) The sanction for correlating a "real" physical 
world to certain feelings of which we are conscious does 
not seem to differ in any essential respect from the 
sanction for correlating a spiritual domain to another 
side of our personality. 

It is not suggested that there is anything new in this 
philosophy. In particular the essence of the first point 
has been urged by many writers, and has no doubt won 
individual assent from many scientists before the recent 
revolutions of physical theory. But it places a somewhat 
different complexion on the matter when this is not 
merely a philosophic doctrine to which intellectual 
assent might be given, but has become part of the 
scientific attitude of the day, illustrated in detail in the 
current scheme of physics. 



CONVICTION 333 

Conviction. Through fourteen chapters you have fol- 
lowed with me the scientific approach to knowledge. 
I have given the philosophical reflections as they have 
naturally arisen from the current scientific conclusions, 
I hope without distorting them for theological ends. In 
the present chapter the standpoint has no longer been 
predominantly scientific; I started from that part of our 
experience which is not within the scope of a scientific 
survey, or at least is such that the methods of physical 
science would miss the significance that we consider it 
essential to attribute to it. The starting-point of belief 
in mystical religion is a conviction of significance or, 
as I have called it earlier, the sanction of a striving in 
the consciousness. This must be emphasised because 
appeal to intuitive conviction of this kind has been the 
foundation of religion through all ages and I do not 
wish to give the impression that we have now found 
something new and more scientific to substitute. I re- 
pudiate the idea of proving the distinctive beliefs of 
religion either from the data of physical science or by 
the methods of physical science. Presupposing a 
mystical religion based not on science but (rightly or 
wrongly) on a self-known experience accepted as fun- 
damental, we can proceed to discuss the various criti- 
cisms which science might bring against it or the 
possible conflict with scientific views of the nature of 
experience equally originating from self-known data. 

It is necessary to examine further the nature of the 
conviction from which religion arises; otherwise we may 
seem to be countenancing a blind rejection of reason as 
a guide to truth. There is a hiatus in reasoning, we must 
admit; but it is scarcely to be described as a rejection 
of reasoning. There is just the same hiatus in reasoning 
about the physical world if we go back far enough. We 



334 SCIENCE AND MYSTICISM 

can only reason from data and the ultimate data must 
be given to us by a non-reasoning process — a self- 
knowledge of that which is in our consciousness. To 
make a start we must be aware of something. But that 
is not sufficient; we must be convinced of the signifi- 
cance of that awareness. We are bound to claim for 
human nature that, either of itself or as inspired by a 
power beyond, it is capable of making legitimate 
judgments of significance. Otherwise we cannot even 
reach a physical world.* 

Accordingly the conviction which we postulate is 
that certain states of awareness in consciousness have 
at least equal significance with those which are called 
sensations. It is perhaps not irrelevant to note that time 
by its dual entry into our minds (p. 51) to some extent 
bridges the gap between sense-impressions and these 
other states of awareness. Amid the latter must be 
found the basis of experience from which a spiritual 
religion arises. The conviction is scarcely a matter to 
be argued about, it is dependent on the forcefulness of 
the feeling of awareness. 

But, it may be said, although we may have such a 
department of consciousness, may we not have mis- 
understood altogether the nature of that which we 
believe we are experiencing? That seems to me to be 
rather beside the point. In regard to our experience of 
the physical world we have very much misunderstood 
the meaning of our sensations. It has been the task of 
science to discover that things are very different from 

* We can of course solve the problem arising from certain data 
without being convinced of the significance of the data — the "official" 
scientific attitude as I have previously called it But a physical world 
which has only the status of the solution of a problem, arbitrarily chosen 
to pass an idle hour, is not what is intended here. 



CONVICTION 335 

what they seem. But we do not pluck out our eyes 
because they persist in deluding us with fanciful 
colourings instead of giving us the plain truth about 
wave-length. It is in the midst of such misrepresenta- 
tions of environment (if you must call them so) that we 
have to live. It is, however, a very one-sided view of 
truth which can find in the glorious colouring of our 
surroundings nothing but misrepresentation — which 
takes the environment to be all important and the 
conscious spirit to be inessential. In our scientific 
chapters we have seen how the mind must be regarded 
as dictating the course of world-building; without it 
there is but formless chaos. It is the aim of physical 
science, so far as its scope extends, to lay bare the 
fundamental structure underlying the world; but science 
has also to explain if it can, or else humbly to accept, 
the fact that from this world have arisen minds capable 
of transmuting the bare structure into the richness of 
our experience. It is not misrepresentation but rather 
achievement — the result perhaps of long ages of bio- 
logical evolution — that we should have fashioned a 
familiar world out of the crude basis. It is a fulfilment 
of the purpose of man's nature. If likewise the spiritual 
world has been transmuted by a religious colour beyond 
anything implied in its bare external qualities, it may 
be allowable to assert with equal conviction that this 
is not misrepresentation but the achievement of a divine 
element in man's nature. 

May I revert again to the analogy of theology with 
the supposed science of humour which (after consulta- 
tion with a classical authority) I venture to christen 
"geloeology". Analogy is not convincing argument, but 
it must serve here. Consider the proverbial Scotchman 
with strong leanings towards philosophy and incapable 



336 SCIENCE AND MYSTICISM 

of seeing a joke. There is no reason why he should not 
take high honours in geloeology, and for example write 
an acute analysis of the differences between British and 
American humour. His comparison of our respective 
jokes would be particularly unbiased and judicial, seeing 
that he is quite incapable of seeing the point of either. 
But it would be useless to consider his views as to which 
was following the right development; for that he would 
need a sympathetic understanding — he would (in the 
phrase appropriate to the other side of my analogy) need 
to be converted. The kind of help and criticism given 
by the geloeologist and the philosophical theologian is 
to secure that there is method in our madness. The 
former may show that our hilarious reception of a 
speech is the result of a satisfactory dinner and a good 
cigar rather than a subtle perception of wit; the latter 
may show that the ecstatic mysticism of the anchorite 
is the vagary of a fevered body and not a transcendent 
revelation. But I do not think we should appeal to 
either of them to discuss the reality of the sense with 
which we claim to be endowed, nor the direction of its 
right development. That is a matter for our inner sense 
of values which we all believe in to some extent, though 
it may be a matter of dispute just how far it goes. If we 
have no such sense then it would seem that not only 
religion, but the physical world and all faith in reasoning 
totter in insecurity. 

I have sometimes been asked whether science cannot 
now furnish an argument which ought to convince any 
reasonable atheist. I could no more ram religious con- 
viction into an atheist than I could ram a joke into the 
Scotchman. The only hope of "converting" the latter 
is that through contact with merry-minded companions 
he may begin to realise that he is missing something 



CONVICTION 337 

in life which is worth attaining. Probably in the recesses 
of his solemn mind there exists inhibited the seed of 
humour, awaiting an awakening by such an impulse. 
The same advice would seem to apply to the propagation 
of religion; it has, I believe, the merit of being entirely 
orthodox advice. 

We cannot pretend to offer proofs. Proof is an idol 
before whom the pure mathematician tortures himself. 
In physics we are generally content to sacrifice before 
the lesser shrine of Plausibility. And even the pure 
mathematician — that stern logician — reluctantly allows 
himself some prejudgments; he is never quite convinced 
that the scheme of mathematics is flawless, and mathe- 
matical logic has undergone revolutions as profound as 
the revolutions of physical theory. We are all alike 
stumblingly pursuing an ideal beyond our reach. In 
science we sometimes have convictions as to the right 
solution of a problem which we cherish but cannot 
justify; we are influenced by some innate sense of the 
fitness of things. So too there may come to us convic- 
tions in the spiritual sphere which our nature bids us 
hold to. I have given an example of one such conviction 
which is rarely if ever disputed — that surrender to the 
mystic influence of a scene of natural beauty is right and 
proper for a human spirit, although it would have been 
deemed an unpardonable eccentricity in the "observer" 
contemplated in earlier chapters. Religious conviction 
is often described in somewhat analogous terms as a 
surrender; it is not to be enforced by argument on those 
who do not feel its claim in their own nature. 

I think it is inevitable that these convictions should 
emphasise a personal aspect of what we are trying to 
grasp. We have to build the spiritual world out of 
symbols taken from our own personality, as we build 



338 SCIENCE AND MYSTICISM 

the scientific world out of the metrical symbols of the 
mathematician. If not, it can only be left ungraspable — 
an environment dimly felt in moments of exaltation but 
lost to us in the sordid routine of life. To turn it into 
more continuous channels we must be able to approach 
the World-Spirit in the midst of our cares and duties in 
that simpler relation of spirit to spirit in which all true 
religion finds expression. 

Mystical Religion. We have seen that the cyclic scheme 
of physics presupposes a background outside the scope 
of its investigations. In this background we must find, 
first, our own personality, and then perhaps a greater 
personality. The idea of a universal Mind or Logos 
would be, I think, a fairly plausible inference from the 
present state of scientific theory; at least it is in harmony 
with it. But if so, all that our inquiry justifies us in assert- 
ing is a purely colourless pantheism. Science cannot tell 
whether the world-spirit is good or evil, and its halting 
argument for the existence of a God might equally well 
be turned into an argument for the existence of a Devil. 
I think that that is an example of the limitation of 
physical schemes that has troubled us before — namely, 
that in all such schemes opposites are represented by 
+ and — . Past and future, cause and effect, are repre- 
sented in this inadequate way. One of the greatest 
puzzles of science is to discover why protons and elec- 
trons are not simply the opposites of one another, 
although our whole conception of electric charge 
requires that positive and negative electricity should be 
related like + and — . The direction of time's arrow 
could only be determined by that incongruous mixture 
of theology and statistics known as the second law of 
thermodynamics; or, to be more explicit, the direction 



MYSTICAL RELIGION 339 

of the arrow could be determined by statistical rules, 
but its significance as a governing fact "making sense 
of the world'* could only be deduced on teleological 
assumptions. If physics cannot determine which way 
up its own world ought to be regarded, there is not much 
hope of guidance from it as to ethical orientation. We 
trust to some inward sense of fitness when we orient the 
physical world with the future on top, and likewise we 
must trust to some inner monitor when we orient the 
spiritual world with the good on top. 

Granted that physical science has limited its scope 
so as to leave a background which we are at liberty to, 
or even invited to, fill with a reality of spiritual import, 
we have yet to face the most difficult criticism from 
science. "Here", says science, "I have left a domain 
in which I shall not interfere. I grant that you have 
some kind of avenue to it through the self-knowledge 
of consciousness, so that it is not necessarily a domain 
of pure agnosticism. But how are you going to deal with 
this domain? Have you any system of inference from 
mystic experience comparable to the system by which 
science develops a knowledge of the outside world? 
I do not insist on your employing my method, which 
I acknowledge is inapplicable; but you ought to have 
some defensible method. The alleged b4sis of experience 
may possibly be valid; but have I any reason to regard 
the religious interpretation currently given to it as 
anything more than muddle-headed romancing?" 

The question is almost beyond my scope. I can only 
acknowledge its pertinency. Although I have chosen the 
lightest task by considering only mystical religion — 
and I have no impulse to defend any other — I am not 
competent to give an answer which shall be anything 
like complete. It is obvious that the insight of con- 



340 SCIENCE AND MYSTICISM 

sciousness, although the only avenue to what I have 
called intimate knowledge of the reality behind the 
symbols of science, is not to be trusted implicitly 
without control. In history religious mysticism has 
often been associated with extravagances that cannot 
be approved. I suppose too that oversensitiveness to 
aesthetic influences may be a sign of a neurotic tem- 
perament unhealthy to the individual. We must allow 
something for the pathological condition of the brain 
in what appear to be moments of exalted insight. One 
begins to fear that after all our faults have been detected 
and removed there will not be any "us" left. But in 
the study of the physical world we have ultimately to 
rely on our sense-organs, although they are capable of 
betraying us by gross illusions; similarly the avenue of 
consciousness into the spiritual world may be beset with 
pitfalls, but that does not necessarily imply that no 
advance is possible. 

A point that must be insisted on is that religion or 
contact with spiritual power if it has any general im- 
portance at all must be a commonplace matter of 
ordinary life, and it should be treated as such in any 
discussion. I hope that you have not interpreted my 
references to mysticism as referring to abnormal experi- 
ences and revelations. I am not qualified to discuss 
what evidential value (if any) may be attached to the 
stranger forms of experience and insight. But in any 
case to suppose that mystical religion is mainly con- 
cerned with these is like supposing that Einstein's 
theory is mainly concerned with the perihelion of 
Mercury and a few other exceptional observations. 
For a matter belonging to daily affairs the tone of 
current discussions often seems quite inappropriately 
pedantic. 



MYSTICAL RELIGION 341 

As scientists we realise that colour is merely a question 
of the wave-lengths of aethereal vibrations; but that does 
not seem to have dispelled the feeling that eyes which 
reflect light near wave-length 4800 are a subject for 
rhapsody whilst those which reflect wave-length 5300 
are left unsung. We have not yet reached the practice of 
the Laputans, who, "if they would, for example, praise the 
beauty of a woman, or any other animal, they describe 
it by rhombs, circles, parallelograms, ellipses, and other 
geometrical terms". The materialist who is convinced 
that all phenomena arise from electrons and quanta and 
the like controlled by mathematical formulae, must 
presumably hold the belief that his wife is a rather 
elaborate differential equation; but he is probably 
tactful enough not to obtrude this opinion in domestic 
life. If this kind of scientific dissection is felt to be 
inadequate and irrelevant in ordinary personal relation- 
ships, it is surely out of place in the most personal 
relationship of all — that of the human soul to a divine 
spirit. 

We are anxious for perfect truth, but it is hard to say 
in what form perfect truth is to be found. I cannot 
quite believe that it has the form typified by an inventory. 
Somehow as part of its perfection there should be in- 
corporated in it that which we esteem as a "sense of 
proportion". The physicist is not conscious of any 
disloyalty to truth on occasions when his sense of 
proportion tells him to regard a plank as continuous 
material, well knowing that it is "really" empty space 
containing sparsely scattered electric charges. And the 
deepest philosophical researches as to the nature of the 
Deity may give a conception equally out of proportion 
for daily life; so that we should rather employ a concep- 
tion that was unfolded nearly two thousand years ago. 



342 SCIENCE AND MYSTICISM 

I am standing on the threshold about to enter a room. 
It is a complicated business. In the first place I must 
shove against an atmosphere pressing with a force of 
fourteen pounds on every square inch of my body. 
I must make sure of landing on a plank travelling at 
twenty miles a second round the sun — a fraction of a 
second too early or too late, the plank would be miles 
away. I must do this whilst hanging from a round 
planet head outward into space, and with a wind of 
aether blowing at no one knows how many miles a 
second through every interstice of my body. The plank 
has no solidity of substance. To step on it is like stepping 
on a swarm of flies. Shall I not slip through? No, if 
I make the venture one of the flies hits me and gives a 
boost up again; I fall again and am knocked upwards 
by another fly; and so on. I may hope that the net result 
will be that I remain about steady; but if unfortunately 
I should slip through the floor or be boosted too vio- 
lently up to the ceiling, the occurrence would be, not 
a violation of the laws of Nature, but a rare coincidence. 
These are some of the minor difficulties. I ought really 
to look at the problem four-dimensionally as concerning 
the intersection of my world-line with that of the plank. 
Then again it is necessary to determine in which direc- 
tion the entropy of the world is increasing in order to 
make sure that my passage over the threshold is an 
entrance, not an exit. 

Verily, it is easier for a camel to pass through the eye 
of a needle than for a scientific man to pass through a 
door. And whether the door be barn door or church 
door it might be wiser that he should consent to be an 
ordinary man and walk in rather than wait till all the 
difficulties involved in a really scientific ingress are 
resolved. 



"1 



CONCLUSION 

A tide of indignation has been surging in the breast of 
the matter-of-fact scientist and is about to be unloosed 
upon us. Let us broadly survey the defence we can set 
up. 

I suppose the most sweeping charge will be that I 
have been talking what at the back of my mind I must 
know is only a well-meaning kind of nonsense. I can 
assure you that there is a scientific part of me that has 
often brought that criticism during some of the later 
chapters. I will not say that I have been half-convinced, 
but at least I have felt a homesickness for the paths of 
physical science where there are more or less discernible 
handrails to keep us from the worst morasses of foolish- 
ness. But however much I may have felt inclined to 
tear up this part of the discussion and confine myself to 
my proper profession of juggling with pointer readings, 
I find myself holding to the main principles. Starting 
from aether, electrons and other physical machinery we 
cannot reach conscious man and render count of what 
is apprehended in his consciousness. Conceivably we 
might reach a human machine interacting by reflexes 
with its environment; but we cannot reach rational man 
morally responsible to pursue the truth as to aether and 
electrons or to religion. Perhaps it may seem unneces- 
sarily portentous to invoke the latest developments of 
the relativity and quantum theories merely to tell you 
this; but that is scarcely the point. We have followed 
these theories because they contain the conceptions of 
modern science; and it is not a question of asserting a 
faith that science must ultimately be reconcilable with 
an idealistic view, but of examining how at the moment 

343 



344 CONCLUSION 

it actually stands in regard to it. I might sacrifice the 
detailed arguments of the last four chapters (perhaps 
marred by dialectic entanglement) if I could otherwise 
convey the significance of the recent change which has 
overtaken scientific ideals. The physicist now regards 
his own external world in a way which I can only describe 
as more mystical, though not less exact and practical, 
than that which prevailed some years ago, when it was 
taken for granted that nothing could be true unless an 
engineer could make a model of it. There was a time 
when the whole combination of self and environment 
which makes up experience seemed likely to pass under 
the dominion of a physics much more iron-bound than 
it is now. That overweening phase, when it was almost 
necessary to ask the permission of physics to call one's 
soul one's own, is past. The change gives rise to 
thoughts which ought to be developed. Even if we 
cannot attain to much clarity of constructive thought 
we can discern that certain assumptions, expectations 
or fears are no longer applicable. 

Is it merely a well-meaning kind of nonsense for a 
physicist to affirm this necessity for an outlook beyond 
physics? It is worse nonsense to deny it. Or as that 
ardent relativist the Red Queen puts it, "You call that 
nonsense, but I've heard nonsense compared with which 
that would be as sensible as a dictionary". 

For if those who hold that there must be a physical 
basis for everything hold that these mystical views are 
nonsense, we may ask — What then is the physical basis 
of nonsense? The "problem of nonsense" touches the 
scientist more nearly than any other moral problem. 
He may regard the distinction of good and evil as too 
remote to bother about; but the distinction of sense 
and nonsense, of valid and invalid reasoning, must be 



CONCLUSION 345 

accepted at the beginning of every scientific inquiry. 
Therefore it may well be chosen for examination as a 
test case. 

If the brain contains a physical basis for the nonsense 
which it thinks, this must be some kind of configuration 
of the entities of physics — not precisely a chemical 
secretion, but not essentially different from that kind 
of product. It is as though when my brain says 7 times 
8 are 56 its machinery is manufacturing sugar, but 
when it says 7 times 8 are 6$ the machinery has gone 
wrong and produced chalk. But who says the machinery 
has gone wrong? As a physical machine the brain has 
acted according to the unbreakable laws of physics; 
so why stigmatise its action? This discrimination of 
chemical products as good or evil has no parallel in 
chemistry. We cannot assimilate laws of thought to 
natural laws; they are laws which ought to be obeyed, 
not laws which must be obeyed; and the physicist must 
accept laws of thought before he accepts natural law. 
"Ought" takes us outside chemistry and physics. It 
concerns something which wants or esteems sugar, not 
chalk, sense, not nonsense. A physical machine cannot 
esteem or want anything; whatever is fed into it it will 
chaw up according to the laws of its physical machinery. 
That which in the physical world shadows the nonsense 
in the mind affords no ground for its condemnation. In a 
world of aether and electrons we might perhaps encounter 
nonsense; we could not encounter damned nonsense. 

The most plausible physical theory of correct rea- 
soning would probably run somewhat as follows. By 
reasoning we are sometimes able to predict events 
afterwards confirmed by observation; the mental pro- 
cesses follow a sequence ending in a conception which 
anticipates a subsequent perception. We may call such 



346 CONCLUSION 

a chain of mental states "successful reasoning" — 
intended as a technical classification without any moral 
implications involving the awkward word "ought". We 
can examine what are the common characteristics of 
various pieces of successful reasoning. If we apply this 
analysis to the mental aspects of the reasoning we obtain 
laws of logic; but presumably the analysis could also 
be applied to the physical constituents of the brain. It 
is not unlikely that a distinctive characteristic would be 
found in the physical processes in the brain-cells which 
accompany successful reasoning, and this would con- 
stitute "the physical basis of success." 

But we do not use reasoning power solely to predict 
observational events, and the question of success (as 
above defined) does not always arise. Nevertheless if 
such reasoning were accompanied by the product which 
I have called "the physical basis of success" we should 
naturally assimilate it to successful reasoning. 

And so if I persuade my materialist opponent to 
withdraw the epithet "damned nonsense" as inconsistent 
with his own principles he is still entitled to allege that 
my brain in evolving these ideas did not contain the 
physical basis of success. As there is some danger of 
our respective points of view becoming mixed up, I 
must make clear my contention: 

(a) If I thought like my opponent I should not worry 
about the alleged absence of a physical basis of success 
in my reasoning, since it is not obvious why this should 
be demanded when we are not dealing with observa- 
tional predictions. 

(b) As I do not think like him, I am deeply perturbed 
by the allegation; because I should consider it to be the 
outward sign that the stronger epithet (which is not 
inconsistent with my principles) is applicable. 



CONCLUSION 347 

I think that the "success" theory of reasoning will 
not be much appreciated by the pure mathematician. 
For him reasoning is a heaven-sent faculty to be enjoyed 
remote from the fuss of external Nature. It is heresy 
to suggest that the status of his demonstrations depends 
on the fact that a physicist now and then succeeds in 
predicting results which accord with observation. Let 
the external world behave as irrationally as it will, there 
will remain undisturbed a corner of knowledge where 
he may happily hunt for the roots of the Riemann- 
Zeta function. The "success" theory naturally justifies 
itself to the physicist. He employs this type of activity 
of the brain because it leads him to what he wants — a 
verifiable prediction as to the external world — and for 
that reason he esteems it. Why should not the theo- 
logian employ and esteem one of the mental processes 
of unreason which leads to what he wants — an assurance 
of future bliss, or a Hell to frighten us into better 
behaviour? Understand that I do not encourage theo- 
logians to despise reason; my point is that they might 
well do so if it had no better justification than the 
"success" theory. 

And so my own concern lest I should have been 
talking nonsense ends in persuading me that I have to 
reckon with something that could not possibly be 
found in the physical world. 

Another charge launched against these lectures may 
be that of admitting some degree of supernaturalism, 
which in the eyes of many is the same thing as super- 
stition. In so far as supernaturalism is associated with 
the denial of strict causality (p. 309) I can only answer 
that that is what the modern scientific development of 
the quantum theory brings us to. But probably the 
more provocative part of our scheme is the role allowed 



348 CONCLUSION 

to mind and consciousness. Yet I suppose that our 
adversary admits consciousness as a fact and he is aware 
that but for knowledge by consciousness scientific 
investigation could not begin. Does he regard con- 
sciousness as supernatural? Then it is he who is 
admitting the supernatural. Or does he regard it as 
part of Nature? So do we. We treat it in what seems 
to be its obvious position as the avenue of approach to 
the reality and significance of the world, as it is the 
avenue of approach to all scientific knowledge of the 
world. Or does he regard consciousness as something 
which unfortunately has to be admitted but which it is 
scarcely polite to mention? Even so we humour him. 
We have associated consciousness with a background 
untouched in the physical survey of the world and have 
given the physicist a domain where he can go round in 
cycles without ever encountering anything to bring a 
blush to his cheek. Here a realm of natural law is 
secured to him covering all that he has ever effectively 
occupied. And indeed it has been quite as much the 
purpose of our discussion to secure such a realm where 
scientific method may work unhindered, as to deal with 
the nature of that part of our experience which lies 
beyond it. This defence of scientific method may not 
be superfluous. The accusation is often made that, by 
its neglect of aspects of human experience evident to 
a wider culture, physical science has been overtaken 
by a kind of madness leading it sadly astray. It is 
part of our contention that there exists a wide field 
of research for which the methods of physics suffice, 
into which the introduction of these other aspects would 
be entirely mischievous. 

A besetting temptation of the scientific apologist for 
religion is to take some of its current expressions and 



CONCLUSION 349 

after clearing away crudities of thought (which must 
necessarily be associated with anything adapted to the 
everyday needs of humanity) to water down the meaning 
until little is left that could possibly be in opposition 
to science or to anything else. If the revised interpre- 
tation had first been presented no one would have raised 
vigorous criticism; on the other hand no one would have 
been stirred to any great spiritual enthusiasm. It is the 
less easy to steer clear of this temptation because it is 
necessarily a question of degree. Clearly if we are to 
extract from the tenets of a hundred different sects any 
coherent view to be defended some at least of them must 
be submitted to a watering-down process. I do not 
know if the reader will acquit me of having succumbed 
to this temptation in the passages where I have touched 
upon religion; but I have tried to make a fight against it. 
Any apparent failure has probably arisen in the following 
way. We have been concerned with the borderland of 
the material and spiritual worlds as approached from the 
side of the former. From this side all that we could 
assert of the spiritual world would be insufficient to 
justify even the palest brand of theology that is not too 
emaciated to have any practical influence on man's 
outlook. But the spiritual world as understood in any 
serious religion is by no means a colourless domain. 
Thus by calling this hinterland of science a spiritual 
world I may seem to have begged a vital question, 
whereas I intended only a provisional identification. To 
make it more than provisional an approach must be made 
from the other side. I am unwilling to play the amateur 
theologian, and examine this approach in detail. I have, 
however, pointed out that the attribution of religious 
colour to the domain must rest on inner conviction; and 
I think we should not deny validity to certain inner 



35o CONCLUSION 

convictions, which seem parallel with the unreasoning 
trust in reason which is at the basis of mathematics, with 
an innate sense of the fitness of things which is at the 
basis of the science of the physical world, and with an 
irresistible sense of incongruity which is at the basis of 
the justification of humour. Or perhaps it is not so much 
a question of asserting the validity of these convictions 
as of recognising their function as an essential part of 
our nature. We do not defend the validity of seeing 
beauty in a natural landscape; we accept with gratitude 
the fact that we are so endowed as to see it that way. 

It will perhaps be said that the conclusion to be 
drawn from these arguments from modern science, is 
that religion first became possible for a reasonable 
scientific man about the year 1927. If we must consider 
that tiresome person, the consistently reasonable man, 
we may point out that not merely religion but most of 
the ordinary aspects of life first became possible for him 
in that year. Certain common activities (e.g. falling in 
love) are, I fancy, still forbidden him. If our expectation 
should prove well founded that 1927 has seen the final 
overthrow of strict causality by Heisenberg, Bohr, Born 
and others, the year will certainly rank as one of the 
greatest epochs in the development of scientific philo- 
sophy. But seeing that before this enlightened era men 
managed to persuade themselves that they had to mould 
their own material future notwithstanding the yoke of 
strict causality, they might well use the same modus 
vivendi in religion. 

This brings us to consider the view often pontifically 
asserted that there can be no conflict between science 
and religion because they belong to altogether different 
realms of thought. The implication is that discussions 
such as we have been pursuing are superfluous. But it 



CONCLUSION 351 

seems to me rather that the assertion challenges this kind 
of discussion — to see how both realms of thought can 
be associated independently with our existence. Having 
seen something of the way in which the scientific realm 
of thought has constituted itself out of a self-closed 
cyclic scheme we are able to give a guarded assent. The 
conflict will not be averted unless both sides confine 
themselves to their proper domain; and a discussion 
which enables us to reach a better understanding as to 
the boundary should be a contribution towards a state 
of peace. There is still plenty of opportunity for frontier 
difficulties; a particular illustration will show this. 

A belief not by any means confined to the more 
dogmatic adherents of religion is that there is a future 
non-material existence in store for us. Heaven is no- 
where in space, but it is in time. (All the meaning of 
the belief is bound up with the word future; there is no 
comfort in an assurance of bliss in some former state of 
existence.) On the other hand the scientist declares that 
time and space are a single continuum, and the modern 
idea of a Heaven in time but not in space is in this 
respect more at variance with science than the pre- 
Copernican idea of a Heaven above our heads. The 
question I am now putting is not whether the theologian 
or the scientist is right, but which is trespassing on the 
domain of the other? Cannot theology dispose of the 
destinies of the human soul in a non-material way 
without trespassing on the realm of science? Cannot 
science assert its conclusions as to the geometry of the 
space-time continuum without trespassing on the realm 
of theology? According to the assertion above science 
and theology can make what mistakes they please 
provided that they make them in their own territory ; they 
cannot quarrel if they keep to their own realms. But 



352 CONCLUSION 

it will require a skilful drawing of the boundary line 
to frustrate the development of a conflict here.* 

The philosophic trend of modern scientific thought 
differs markedly from the views of thirty years ago. 
Can we guarantee that the next thirty years will not see 
another revolution, perhaps even a complete reaction? 
We may certainly expect great changes, and by that 
time many things will appear in a new aspect. That is 
one of the difficulties in the relations of science and 
philosophy; that is why the scientist as a rule pays so 
little heed to the philosophical implications of his own 
discoveries. By dogged endeavour he is slowly and 
tortuously advancing to purer and purer truth; but his 
ideas seem to zigzag in a manner most disconcerting 
to the onlooker. Scientific discovery is like the fitting 
together of the pieces of a great jig-saw puzzle; a 
revolution of science does not mean that the pieces 
already arranged and interlocked have to be dispersed; 
it means that in fitting on fresh pieces we have had to 
revise our impression of what the puzzle-picture is 
going to be like. One day you ask the scientist how he is 
getting on; he replies, "Finely. I have very nearly 
finished this piece of blue sky." Another day you ask 
how the sky is progressing and are told, "I have added a 
lot more, but it was sea, not sky; there's a boat floating on 
the top of it". Perhaps next time it will have turned out 
to be a parasol upside down ; but our friend is still enthusi- 
astically delighted with the progress he is making. The 
scientist has his guesses as to how the finished picture will 
work out; he depends largely on these in his search for 
other pieces to fit; but his guesses are modified from time 
to time by unexpected developments as the fitting pro- 

*This difficulty is evidently connected with the dual entry of time 
into our experience to which I have so often referred. 



CONCLUSION 353 

ceeds. These revolutions of thought as to the final 
picture do not cause the scientist to lose faith in his 
handiwork, for he is aware that the completed portion 
is growing steadily. Those who look over his shoulder 
and use the present partially developed picture for 
purposes outside science, do so at their own risk. 

The lack of finality of scientific theories would be a 
very serious limitation of our argument, if we had staked 
much on their permanence. The religious reader may 
well be content that I have not offered him a God 
revealed by the quantum theory, and therefore liable 
to be swept away in the next scientific revolution. It is 
not so much the particular form that scientific theories 
have now taken — the conclusions which we believe we 
have proved — as the movement of thought behind them 
that concerns the philosopher. Our eyes once opened, 
we may pass on to a yet newer outlook on the world, 
but we can never go back to the old outlook. 

If the scheme of philosophy which we now rear on 
the scientific advances of Einstein, Bohr, Rutherford 
and others is doomed to fall in the next thirty years, it 
is not to be laid to their charge that we have gone astray. 
Like the systems of Euclid, of Ptolemy, of Newton, 
which have served their turn, so the systems of Einstein 
and Heisenberg may give way to some fuller realisation 
of the world. But in each revolution of scientific thought 
new words are set to the old music, and that which has 
gone before is not destroyed T^ut refocussed. Amid all 
our faulty attempts at expression the kernel of scientific 
truth steadily grows; and of this truth it may be said — 
The more it changes, the more it remains the same 
thing. 



INDEX 



A B C of physics, xiv, 88 

A priori probability, 77, 244, 305 

Absolute, 23, 56; past and future, 

48, 57, 295; elsewhere, 49, 50; 

values, 288, 331; future perfect, 

307 
Absorption of light, 184, 186 
Abstractions, 53 

Accelerated frames of reference, 113 
Acceleration, relativity of, 129 
Action, 180, 241; atom of, 182 
Actuality, 266, 319 
Aether, nature of, 31 
Aether-drag, 3 
Age of the sun, 169 
And, study of, 104 
Anthropomorphic conception of 

deity, 282, 337, 341 
Antisymmetrical properties of 

world, 236 
Ape-like ancestors, 16, 81, 273 
Apple (Newton's), hi, 115 
Arrow, Time's, 69, 79, 88, 295 
Astronomer Royal's time, 36, 40 
Atom, structure of, 1, 190, 199, 224 
Atom of action, 182. See Quantum 
Atomicity, laws of, 236, 245 
Averages, 300 
Awareness, 16, 334 

Background of pointer readings, 

137, 255, 259, 268, 330, 339 
Balance sheet, 33 
Beats, 216 

Beauty, 105, 267, 350 
Becoming, 68, 87 
Beginning of time, 83 
Berkeley, Bishop, xii, 326 



Beta (3) particle, 59 

Bifurcation of the world, 236 

Billiard ball atoms, 2, 259 

Blessed gods (Hegel), 147, 155 

Bohr, N., 2, 185, 191, 196, 220, 306 

Boltzmann, L., 63 

Bombardment, molecular, 113, 131 

Born, M., 208 

Bose, S. N., 203 

Bragg, W. H., 194 

Brain, 260, 268, 279, 311, 323 

Broad, C. D., 160 

de Broglie, L., 201, 202 

Building material, 230 

Bursar, 237 

Casual and essential characteristics, 

142 
Categories, xi, 105 
Causality, 297 
Cause and effect, 295 
Cepheid variables, 165 
Chalk, calculation of motion of, 107 
Chance, 72, 77, 189 
Classical laws and quantum laws, 

193, 195, 308 
Classical physics, 4 
Clifford, W. K., 278 
Clock, 99, 134, 154 
Code-numbers, 55, 81, 235, 277 
Coincidences, 71 
Collection-box theory, 187, 193 
Colour and wave-length, 88, 94, 

329, 34i 
Commonsense knowledge, 16 

Companion of Sirius, 203 

Comparability of relations, 232 

Compensation of errors, 12 



355 



356 



INDEX 



Concrete, 273 

Configuration space, 219 

Conservation, laws of, 236, 241 

Constellations, subjectivity of, 95, 
106, 241 

Contiguous relations, 233 

Contraction, FitzGerald, 5, 24; 
reality of, 32, 53 

Controlling laws, 151, 245 

Conversion, 336 

Conviction, 333, 350 

Co-ordinates, 208, 231 

Copenhagen school, 195 

Correspondence principle, 196 

Counts of stars, 163 

Crudeness of scale and clock sur- 
vey, 154 

Curvature of space-time, 119, 127, 
157; coefficients of, 120, 155 

Cyclic method of physics, 260, 277, 

Cylindrical curvature, 139 

Darwin, G. H., 171 

Deflection of light by gravity, 122 

Demon (gravitation), 118, 309 

Dense matter, 203 

Design, 77 

Detailed balancing, principle of, 80 

Determinism, 228, 271, 294, 303, 310 

Differential equations, 282, 329, 341 

Diffraction of electrons, 202 

Dimension, fourth, 52; beyond 
fourth, 120, 158, 219 

Dirac, P. A. M., 208, 219, 270 

Directed radius, 140 

Direction, relativity of, 26 

Distance, relativity of, 25; inscru- 
table nature of, 81; macroscop- 
ic character, 155, 201 

Door, scientific ingress through, 342 

Doppler effect, 45, 184 

Double stars, 175 

Dual recognition of time, 51, 91, 99, 
334, 352 



Duration and becoming, 79, 99 
Dynamic quality of time, 68, 90, 92, 
260 

Eclipses, prediction of, 149, 299 

Ego, 97, 282, 315 

Egocentric attitude of observer, 15, 
61, 112 

Einstein, A., 1, 53, in, 185, 203 

Einstein's law of gravitation, 120, 
J 39» I 5 I > 260; law of motion, 
124 

Einstein's theory, 20, 111 

Electrical theory of matter, 2, 6 

Electromagnetism, 236 

Electron, 3 ; mass of, 59 ; extension 
in time, 146; in the atom, 188, 
199, 224; nature of, 279, 290 

Elephant, problem of, 251 

Elliptical space, 289 

Elsewhere, 42 

Emission of light, 183, 191, 216 

Encounters of stars, 177 

Engineer, superseded by mathe- 
matician, 104, 209 

Entropy, 74, 105 

Entropy-change and Becoming, 88 

Entropy-clock, 101 

Environment, 288, 328 

Epistemology, 225, 304 

Erg-seconds, 179 

Essential characteristics, 142 

Euclidean geometry, 159 

Events, location of, 41 ; point- 
events, 49 

Evolution, irreversibility of, 91 ; in 
stellar system, 167, 176 

Exact science, 250 

Existence, 286 

Experience, 288, 328 

Explanation, scientific ideal of, xiii, 
138, 209, 248 

Extensive abstraction, method of, 
249 

External world, 284 



INDEX 



357 



Familiar and scientific worlds, xiii, 
247, 324 

Fictitious lengths, 19 

Field, 153 

Field-physics, 236 

Finite but unbounded space, 80, 139, 
166, 289 

FitzGerald contraction, 5, 24; real- 
ity of, 32, 53 

Flat world, 118, 138 

Flatness of galaxy, 164 

Force, 124 

Formality of taking place, 68 

Fortuitous concourse of atoms, 77, 
251 

Fourth dimension, 52, 231 

Fowler, R. H., 204 

Frames of space and time, 14, 20, 
35, 61, 112, 155 

Freak (solar system), 176 

Freewill, 295 

Fullness of space, measures of, 153 

Future, relative and absolute, 48 ; 
see Predictability 

Future life, 351 

Future perfect tense, 307 

Galactic system, 163 

Geloeology, 335 

General theory of relativity, in, 
129 

Generation of Waves by Wind, 
316 

Geodesic, 125 

Geometrisation of physics, 136 

Geometry, 133, 157, 161 

Grain of the world, 48, 55, 56, 90 

Gravitation, relative and absolute 
features, 114; as curvature, 
118; law of, 120, 139; explana- 
tion of, 138, 145 

Greenland, 117 

Gross appliances, survey with, 154 

Growth, idea of, 87 

Group velocity, 213 



h, 179, 183, 223 

Halo of reality, 282, 285, 290 

Hamilton, W. R., 181 

Hamiltonian differentiation, 240 

Heaven, 351 

Hegel, 147 

Heisenberg, W., 206, 220, 228, 306 

Heredity, 250 

Here-Now, 41 

Heterodyning, 216 

Hour-glass figures, 48 

House that Jack Built, 262 

Hubble, E. P., 167 

Humour, 322, 335 

Humpty Dumpty, 64 

Huxley, T. H., 173 

Hydrodynamics, 242, 316 

Hydrogen, 3 

Hyperbolic geometry, 136 

Hypersphere, 81, 157 

i (square root of — 1), 135, 146, 

208 
Identical laws, 237 
Identity replacing causation, 156 
Illusion, 320 

Impossibility and improbability, 75 
Impressionist scheme of physics, 103 
Indeterminacy, principle of, 220, 

306 
Inertia, 124 

Inference, chain of, 270, 298 
Infinity, 80 

Infra-red photography, 173 
Inner Light, 327 

Insight, 89, 91, 268, 277, 311, 339 
Instants, world-wide, 43 
Integers, 220, 246 
Interval, 37, 261 
Intimate and symbolic knowledge, 

321 
Introspection, 321 
Invariants, 23 
Inventory method, 103, 106, 280, 

341 



358 



INDEX 



Inverse-square law, 29 

Island universes, 165 

Isotropic directed curvature, 144 

Jabberwocky, 291 
Jeans, J. H., 176, 187 
Johnson, Dr., 326 
Jordan, P., 208 

Knowable to mind, 264 
Knowledge, nature of physical, 257, 
304; complete, 226 

Laplace, 176 

Laputans, 341 

Larmor, J., 7 

Laws of Nature, 237, 244 

Laws of thought, 345 

Lenard, P., 130 

Length, 6, 160. See Distance 

Life on other planets, 170 

Life-insurance, 300 

Lift, man in the, 111 

Light, velocity of, 46, 54; emission 

of, 183, 191, 216 
Likeness between relations, 232 
Limitations of physical knowledge, 

257 
Linkage of scientific and familiar 

worlds, xiii, 88, 156, 239, 249 
Location, frames of, 14, 41 
Logos, 338 
Longest track, law of, 125, 135, 

148 
Lorentz, H. A., 7 
Lowell, P., 174 
Luck, rays of, 190 
Lumber (in world building), 235, 

243 

Macroscopic survey, 154, 227, 299, 

304 
Man, 169, 178 
Man-years, 180 
Mars, 172 



Mass, increase with velocity, 39, 50, 

59 

Mathematician, 161, 209, 337, 347 

Matrix, 208 

Matter, 1, 31, 156, 203, 248, 262 

Maxwell, J. C, 8, 60, 156, 237 

Measures of structure, 234, 268 

Mechanical models, 209 

Mechanics and Geometry, 137 

Mendelian theory, 250 

Mental state, 279 

Metric, 142, 153 

Metrical and non-metrical proper- 
ties, 275 

Michelson-Morley experiment, 5, zi 

Microscopic analysis, reaction from, 
103 

Milky Way, Z63 

Miller, D. C, 5 

Mind and matter, 259, 268, 278; 
selection by mind, 239, 243, 264 

Mind-stuff, 276 

Minkowski, H., 34, 53 

Mirror, distortion by moving, it 

Models, 198, 209, 344 

Molecular bombardment, 113, Z3Z 

Momentum, 153, 208, 223, 239, 262 

Monomarks, 23 z 

Moon, origin of, Z7Z 

Morley, E. W., 5 

Motion, law of, 123 

Multiplicationist, 86 

Multiplicity of space and time 
frames, 20, 35, 6z 

Myself, 42, 53 

Mysticism, defence of, 323 ; reli- 
gious, 338 

Nautical Almanac, Z50 

Nebulae, 165 

Nebular observers, 9, Z2 

Neptune, 49 

Neutral stuff, 280 

Neutral wedge, 48 

New quantum theory, 206 



INDEX 



359 



Newton, in, 122, 201; quotation 

from, in 
Newtonian scheme, 4, 18, 125 
Non-empty space, 127, 153, 238 
Non-Euclidean geometry, 157 
Nonsense, problem of, 344 
Now-lines, 42, 47, 49, 184 
Nucleus of atom, 3 

Objectivity of "becoming", 94; of 

a picture, 107 
Observer, attributes of, 15, 337 
Odds, 301, 303 

Official scientific attitude, 286, 334 
Operator, 208 
Orbit jumps of electron, 191, 196, 

205, 215, 300, 312 
Organisation, 68, 70, 104 
Ought, 345 
Oxygen and vegetation, 174 

/»'s and q's, 208, 223, 327 

Pacific Ocean, 171 

Particle, 202, 211, 218 

Past, relative and absolute, 48 

Pedantry, 340, 342 

Permanence, 241 

Personal aspect of spiritual world, 

337 
Phoenix complex, 85 

Photoelectric effect, 187 

Photon, 190 

Physical time, 40 

Picture and paint, 106 

Picture of gravitation, 115, 138, 157 

Plan, Nature's, 27 

Planck, M., 185 

Plurality of worlds, 169 

Pointer readings, 251 

Ponderomotive force, 237 

Porosity of matter, 1 

Potential (gravitational), 261 

Potential energy, 213 

Potential gradient, 96 

Pound sterling, relativity of, 26 



Predestination, 293, 303 

Predictability of events, 147, 228, 
300, 307 

Primary law, 66, 75, 98 ; insuffi- 
ciency of, 107 

Primary scheme of physics, 76, 129, 
295 

Principal curvature, 120, 139 

Principia, 4 

Principle, Correspondence, 196 

Principle of detailed balancing, 80 

Principle of indeterminacy, 220, 306 

Probability, 216, 315 

Proof and plausibility, 337 

Proper-distance, 25 

Proper-time, 37 

Proportion, sense of, 341 

Proton, 3 

Psi {\p), 216, 305 

Pure mathematician, 161, 337, 347 

Purpose, 105 

g-numbers, 208, 270 
Quantum, 184; size of, 200 
Quantum laws, 193 
Quantum numbers, 191, 205 
Quest of the absolute, 26, 122; of 
science, no, 287; of reality, 328 
Quotations from 

Boswell, 326 

Brooke, Rupert, 317 

Clifford, W. K., 278 

Dickens, 32 

Einstein, A., 294 

Hegel, 147 

Huxley, T. H., 173 

Kronecker, L., 246 

Lamb, H., 316 

Lewis Carroll, 28, 291, 344 

Milton, 167 

Newton, in 

Nursery Rhymes, 64, 70, 262 

Omar Khayyam, 64, 293 

O'Shaughnessy, A., 325 

Russell, Bertrand, 160, 278 



360 



INDEX 



Quotations from (cont.) 

Shakespeare, 21, 39, 83, 292, 330 
Swift, 341 
Whitehead, A. N., 145 

Radiation pressure, 58 

Random element, 64; measurement 

of, 74 

Reality, meaning of, 282, 326 

Really true, 34 

Rectification of curves, 125 

Rejuvenescence, theories of, 85, 169 

Relata and relations, 230 

Relativity of velocity, 10, 54, 59, 61 ; 
of space-frames, 21 ; of mag- 
netic field, 22 ; of distance, 25 ; 
of pound sterling, 26; of Now 
(simultaneity), 46, 61; of ac- 
celeration, 129; of standard of 
length, 143 

Religion, 194, 281, 288, 322, 324, 

326, 333, 349 
Retrospective symbols, 307, 308 
Revolutions of scientific thought, 4, 

352 
Right frames of space, 18, 20 

Roemer, O., 43 

Rotating masses, break-up of, 176 

Running down of universe, 63, 84 

Russell, B., 160, 277, 278 

Rutherford, E., 2, 327 

Scale (measuring), 12, 18, 24, 134, 

141 
Schrodinger's theory, 199, 210, 225, 

305 
Scientific and familiar worlds, xiii, 

247, 324 
Second law of thermodynamics, 74, 

86 

Secondary law, 75, 79, 98 

Seen-now line9, 44, 47 

Selection by mind, 239, 243, 264, 

330 
Self-comparison of space, 145 



Sense-organs, 51, 96, 266, 329 

Shadows, world of, xiv, 109 

Shuffling, 63, 92, 184 

Sidereal universe, 163 

Signals, speed of, 57 

Significances, 108, 329 

Simultaneity, 49, 61 

Singularities, 127 

Sirius, Companion of, 203 

de Sitter, W., 167 

Slithy toves, 291 

Solar system, origin of, 176 

Solar system type of atom, 2, 190 

Sorting, 93 

Space, 14, 16, 51, 81, 137 

Spasmodic moon, 226 

Spatial relations, 50 

Spectral lines, 205, 216; displace- 
ment of, 121, 166 

Spherical curvature, radius of, 
14c 

Spherical space, 82, 166, 289; ra- 
dius of, 167 

Spiral nebulae, 165 

Spiritual world, 281, 288, 324, 349 

Standard metre, 141 

Stars, number of, 163 ; double, 175 ; 
evolution of, 176; white 
dwarfs, 203 

States, 197, 301 

Statistical laws, 244; mind's inter- 
ference with, 313 

Statistics, 201, 300, 303 

Stratification, 47 

Stress, 129, 155, 262 

Structure, 234, 277 

Sub-aether, 211, 219 

Subjective element in physics, 95, 
241 

Substance, ix, 273, 318 

Success, physical basis of, 346 

Sun, as a star, 164; age of, 169 

Supernatural, 309, 348 

Survey from within, 145, 321, 330 

Sweepstake theory, 189 



INDEX 



361 



Symbolism in science, xiii, 209, 247, 

269, 324 
Synthetic method of physics, 249 

Temperature, 71 

Temporal relations, 50 

Tensor, 257 

Tensor calculus, 181 

Thermodynamical equilibrium, 77 

Thermodynamics, second law of, 
66, 74, 86 

Thermometer as entropy-clock, 99, 
101 

Thinking machine, 259 

Thought, 258 ; laws of, 345 

Time in physics, 36; time lived 
(proper-time), 40; dual recog- 
nition of, 51, 100; time's arrow, 
69; infinity of, 83; summary of 
conclusions, 101 ; time-triangies, 
133 ; reality of, 275 

Time-scale in astronomy, 167 

Touch, sense of, 273 

Track, longest, 125, 135, 148 

Trade Union of matter, 126 

Transcendental laws, 245 

Traveller, time lived by, 39, 126, 

135 

Triangles in space and time, 133 

Tug of gravitation, 115, 122 

Undoing, 65 
Unhappening, 94, 108 
Uniformity, basis of, 145 



Unknowable entities, 221, 308 
Utopia, 265 

Values, 243, 330 
Vegetation on Mars, 173 
Velocity, relativity of, 10; upper 

limit to, 56 
Velocity through aether, 30, 32 
Velocity of light, 46, 54 
Venus, 170 
Victorian physicist, ideals of, 209, 

259 
View-point, 92, 283 
Void, 13, 137 
Volition, 310 

Watertight compartments, 194 

Wave-group, 213, 217, 225 

Wave-length, measurement of, 24 

Wave-mechanics, 211 

Wave-theory of matter, 202 

Wavicle, 201 

Wells, H. G., 67 

White dwarfs, 203 

Whitehead, A. N., 145, 249 

Whittaker, E. T., 181 

Winding up of universe, 83 

World building, 230 

World-lines, 253 

Worm, four-dimensional, 42, 87, 92 

Wright, W. H., 172 

Wrong frames of reference, 116 

X (Mr.), 262, 268 



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